- Write short notes on the following types of probability distributions:

a] Poisson probability distribution

b] Uniform probability distribution

c] Exponential probability distribution

2] Six(6) doctors and twenty(20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

3] Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8^{th} relationship.

4} To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using Lambda (the mean) to the nearest whole number. What is the probability that it contains:

a] No chocolate chips?

b} 7 chocolate chips?

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value in a list Examples include number of mosquito bite on person, number of computer crashes in a day etc.

Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) ×/0

=0.0067×1/1

=0.0067

P(x=7) ×/5040

= 0.0067/5040

=523.4/5040

=0.1038.

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value in a list Examples include number of mosquito bite on person, number of computer crashes in a day etc.

Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65⁷

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) 2.7183–⁵×5⁰/0

=0.0067×1/1

=0.0067

P(x=7) 2.7183⁷×5⁷/5040

= 0.0067×78125/5040

=523.4/5040

=0.1038.

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution:This is a probability distribution that was named after a French mathematician named Simeon Denis Poission. it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value eg.1,2,3 etc with no fractions or decimals

Uniform probability distribution: it refers to a type of probability distribution with equally like outcomes.eg when a coin is tossed, the probability of getting either a head or tail is the same.There are two types namely; discrete and continous probability distribution.

Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution.

Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.0260.

Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 〖0.65〗^7

0.35×0.0490

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

Lamba=50÷12=4.15

=4(nearest whole number)

P(x=0) 〖2.7183〗^(-4)×4^0/0

=0.0183×1/1

=0.0183

P(x=7) 〖2.7183〗^(-4)×4□7/5040

= 0.0183×16384/5040

=299.82/5040

=0.0595

Name: Chibueze Manna Chioma

Reg No: 2019/244094

Email: chibuezemanna@yahoo.com

1. i. Poisson Probability Distribution

This is a discrete probability that studies the averages of events and experiments. It shows the number or average an event occurs in a specific or fixed period of time.

Formula:

P(X=x)= e–λ λx/ x!

Where: e= exponential (2.17183)

λ= mean

ii. Uniform Probability Distribution

This is a continuous distribution in which events from random experiments have same and equal probability. It is a distribution with likely equally outcomes.

Formula:

P(X=x)= 1/b–a

Where: a= minimum value of x

b= maximum value of x

iii. Exponential Probability Distribution

This is a continuous probability distribution used to measure the length of time of an experiment or event. It is to measure the amount of time or the average amount of time in an event.

Formula:

P(X=x)= me–mx

Where: m= 1/μ; μ= mean

e= exponential (2.17183)

Therefore;

P(X=x)= 1/μ(e–1/μ(x))

2. Let Doctors be D and nurses be N.

D=6

N=20

Without replacement

AND represents Multiplication

P(D)= 4/6

P(N)= 3/20

P(D)× P(N)

= 4/6 × 3/20= 1/10

CONCLUSION: The probability of having 4 doctors and 3 nurses is 1/10.

3. Geometric Distribution

Formula: pqX

Where: p= probability (success)

q= (1-p) (failure)

x= number of failed selections (x-1)

Then, p=0.35

q= (1-0.35= 0.65)

x= 8-1= 7

=(0.35)× (0.65)7

=0.35× 0.04902227891

P(x=8)= 0.01715779762

4. Poisson Distribution

Formula: e-λ λx/x!

Where: λ= mean

λ= 50/12= 4.1667777777

To the nearest whole number= 4.

a. P(x=0)

=e-4 × 4⁰/0!

=e–⁴×1/1= 0.01831563889

P(x=0)= 0.018315637889

b. P(x=7)

=e–⁴× 4⁷/7!

=e–⁴× 16384/5040

=0.05954036261

P(x=7)= 0.05954036261

1a.poisson probability distribution is a type of discrete probability distribution that is used to show how many times an event is likely to occur over a specific period. It’s often used to understand independent events that occur at a constant rate within a given interval of time. It’s also used in testing how unusual an event frequency is for a given interval. It’s extremely helpful for planning purposes as it enables managers to analyze customer behaviour as they visit a restaurant or store. Where we are going to use √ as our lamba

The formula is P(X=x)=e^-√×√^x÷x!

Where x=number of events in an area of opportunity

√=expected number of events (which is used here as lamba)

e=base of the natural logarithm system (2.71828……)

Mean E(x)=√

Variance =√

Standard deviation=square root of √

B. Uniform probability distribution is a type of continuous probability distribution in which all outcomes are equally likely. It’s sometimes also known as a rectangular distribution, is a distribution that has a constant probability. It’s used in situations in which every outcome in a sample space is equally likely

The formula = X~U(a,b)

a=lowest value of x

b= highest value of x

C. Exponential probability distribution: this is a type of continuous probability distribution which is probability of time between events in a poisson point process i.e a process in which events occur continuously and independently at a constant average rate. It’s present when we are dealing with events that are rapidly changing early on for example: news, blog

The formula=P(X=x) or p(x)

2. N= 6+20 =26 ,n=7, x= 4 doctors and 3 nurses, r=6.

(r of x) (N – r, n- x) ÷(N of n)

=(6 of 4) (26 -6, 7-4) ÷(26 of 7)

=(6C4) (20C3)÷(26C7)

15×1140÷657800

=17100÷657800

= 0.025995743 =0.025996

3. p= 0.35

q=1-p which is 1-0.35=0.65

x=7

pq^x = 0.35×0.65^7

=0.017157797=0.017158

4. e^-√ ×√^x÷x!,, Where this(√)is our lamba

When x=0

e^-50×50^0÷0!

=e^-50×1÷1

=1.928749848×10^-22

=1.9287×10^-22

When x=7

=e^-50×50^7÷7!

=e^-50×50^7÷5040

=2.989753609×10^-14

=2.9896×10^-14

Poisson probability distribution is a probability that is used to show how many times an event is likely to occur over a period of time.it’s used in testing how unusual an event frequency is for a given interval.its helpful for planning purposes.

Formula:P(x; μ) = (e-μ) (μx) / x!.

Where x=number of event

μ=expected number of events

e=Base of natural logarithm system(2.71828)

Uniform probability distribution is a continuous probability distribution with all outcomes being equally likely,it’s also known as rectangular distribution.its a distribution which has a constant probability.

Formula:X~U(a,b)

a=lowest value of X

b=highest value of X

Exponential probability distribution is usually present when we are dealing with events that are rapidly changing.its the distribution of the time between events in a poisson point process.this means that it’s a process in which events occur continuously and independently at a constant average rate.

Formula:P(X=c) or P(x)

2: (rCx) (N-rCn-x) /(NCn

N=6+20=26

n=7

x=4doctors

r=6

(6C4) (26-6 C 6-4)/(26C7)

=(6C4) (20C3)/(26C7)

=15*1140 /657800 =17100/657800

=0.025996

3:p=0.35

q=0.65

x=7

Pq^x=0.35*0.65^7

=0.01716

4:p=50

x=0

e- λ . λ^x / x!

e-50 .50^0 /0!

=e-50 .1 /1

=1.9288

x=7

e-50 . 50^7 /7!

e-50 . 50^7 /5040

=2.9895

Write short notes on the following types of probability distributions:

POISSON DISTRIBUTION: This is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here there can be unlimited number of possible outcomes unlike in Binomial Distribution where we have only two possible outcomes.

UNIFORM DISTRIBUTION: It’s a term used to describe a form of probability distribution where every possible outcomes has an equal likelihood of happening, The probability is constant since each variable has equal chances of being the outcome.

EXPONENTIAL DISTRIBUTION: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

2). Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

= (6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3). Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will get married in her 8th relationship?

P(X = x) Pqx

P= 0.35

Q = 1 – p

= 1 – 0.35

= 0.65

x = x – 1

= 8 – 1

= 7.

P(x=7) = 0.35 × 〖0.65〗^7

= 0.0176.

4). To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using landa to the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) 〖2.7183〗^(-5)×5^0/0

=0.0067×1/1

=0.0067

P(x=7) 〖2.7183〗^(-5)×5^7/5040

= 0.0067×78125/5040

=523.4/5040

=0.1038.

Write short notes on the following types of probability distributions:

POISSON DISTRIBUTION: This is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here there can be unlimited number of possible outcomes unlike in Binomial Distribution where we have only two possible outcomes.

UNIFORM DISTRIBUTION: It’s a term used to describe a form of probability distribution where every possible outcomes has an equal likelihood of happening, The probability is constant since each variable has equal chances of being the outcome.

EXPONENTIAL DISTRIBUTION: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

2). Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

= (6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3). Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will get married in her 8th relationship?

P(X = x) Pqx

P= 0.35

Q = 1 – p

= 1 – 0.35

= 0.65

x = x – 1

= 8 – 1

= 7.

P(x=7) = 0.35 × 〖0.65〗^7

= 0.0176.

4). To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using landa to the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) 〖2.7183〗^(-5)×5^0/0

=0.0067×1/1

=0.0067

P(x=7) 〖2.7183〗^(-5)×5^7/5040

= 0.0067×78125/5040

=523.4/5040

=0.1038.

A. POISSON PROBABILTY DISTRIBUTION

This is used to show how many times an event is likely to occur over a specified period. It is known as a count distribution. It gives us the probability of a given number of events happening in a fixed interval of time. Here, there can be unlimited number of possible outcomes unlike in binomial where there is only two possible outcomes(i.e success and failure).

B.UNIFORM PROBABILITY DISTRIBUTION:

Uniform distribution is a continuous probability distribution and it is concerned with events that are equally likely to occur. A distribution in which all outcomes are equally likely.

C. EXPONENTIAL PROBABILITY DISTRIBUTION:

The distribution is also known as negative exponential distribution, it describes time between events in a Poisson process. It is mostly used for testing product reliability. The exponential often models waiting times and can help you to answer questions like: How much time will i go before my car breaks down?

2. Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

4. To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a. No chocolate

b. 7 chocolates.

a. P(x=0) ×/0

=0.0067×1/1

=0.0067

b. P(x=7) ×/5040

= 0.0067/5040

=523.4/5040

=0.1038.

NAME: SIBEUDU CHUKWUEBUKA RALUCHUKWU

REG. NO: 2019/244735

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The probability density function is given as;

{ P(X=x) = (λ^x*e^(-λ)) / x! }

where, e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of x when that is also equal to its variance

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. The probability density function is given as:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

The mean = 0

The variance = 1

The standard deviation = 1

c) Exponential probability distribution:

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes, it is found in various other contexts. The probability density function is given as;

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

Before she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

1a. POISSON PROBABLITY DISTRIBUTION

The distribution is called the Poisson distribution (after S.D Poisson who discovered it in the early part of the nineteenth century),a random variable having this distribution is said to be Poisson distributed. It often appears in connection with the study of sequence of random events occurring over time or space. It has been used to model the distribution of the number of events, in a specific period of time or a specific unit of space (e.g the number of customers passing through a cashier’s counter ). This distribution is very important to managers.

b. UNIFORM PROBABILITY DISTRIBUTION

Uniform probability distribution is also called a rectangular distribution. It plays a very important role in statistics and econometrics. It has a special use in rounding errors when measurements are recorded to some degree of accuracy.

c. EXPONENTIAL PROBABILITY DISTRIBUTION

Exponential distribution is another special case of the Gamma distribution. It is the only continuous distribution with memory less property. It is of considerable importance and widely used in statistics and econometrics. It has been used to model time duration of economic events such as the employment spell of a worker. It is often used in practical problem or represent the distribution of time that elapse before the occurrence of some events, for instance, the time required to serving a customer at some service facility.

2. P (X=4 doctors and 3 nurses)

N = 6+20=26, n=7,r=6

P (X=x) = [rCx × N-rCn-x] ÷ [NCn]

P (X=4 doctors and 3 nurses)=[ 6C4 × 20C3]÷[ 26C7]

= [15 × 1140] ÷ [657,800]

= 0.0259

3. P=0.35

q =1-p = 0.65

x=7

P(x=7) = (0.35)×(0.65)^7

= 0.0172

4.a e=2.7183, ∆=50/12=4, x=0

P(x=0)= [e^-4(4)0]÷0!

=0.0183

4b. P(x=7)= [e^-4(4)^7]=7!

= 0.0595

NAME: ADIGWE ANTHONY CHIBUIKEM

REG. NO: 2019/245463

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time from now until an earthquake occurs has an exponential distribution. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The formula is stated below:

{ f(x)= λe^(- λx )}

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

Before she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: ONYISHI CYNTHIA CHETACHI

REG. NO: 2019/243107

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

In the context of probability distributions, uniform distribution refers to a probability distribution for which all of the values that a random variable can take occur with equal probability. The probability distribution is defined as follows:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. An interesting property of the exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution. The formula is:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

Before she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: EZEUGWU CHIDERA PAUL

REG. NO: 2019/241560

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

In the context of probability distributions, uniform distribution refers to a probability distribution for which all of the values that a random variable can take occur with equal probability. The probability distribution is defined as follows:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. An interesting property of the exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution. The formula is:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^xBefore she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^-4 ] / 0!

= [ 1 * (2.7183)^-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^-4 ] / 7!

= [ 16384 * (2.7183)^-4 ] / 5040

= 0.05953877054

NAME: OGBONNA SANDRA CHINENYE

REG. NO: 2019/245659

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

In the context of probability distributions, uniform distribution refers to a probability distribution for which all of the values that a random variable can take occur with equal probability. The probability distribution is defined as follows:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. An interesting property of the exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution. The formula is:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^xBefore she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^-4 ] / 0!

= [ 1 * (2.7183)^-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^-4 ] / 7!

= [ 16384 * (2.7183)^-4 ] / 5040

= 0.05953877054

NAME: UGWUOKE KOSISOCHUKWU PRECIOUS

REG. NO: 2019/243547

DEPARTMENT: ECONOMICS

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^-4 ] / 0!

= [ 1 * (2.7183)^-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^-4 ] / 7!

= [ 16384 * (2.7183)^-4 ] / 5040

= 0.05953877054

NAME: ANYANWOR CHINENYE FRANCES

REG. NO: 2019/244250

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

A Poisson distribution can be used to analyze the probability of various events regarding how many customers go through the drive-through. It can allow one to calculate the probability of a lull in activity (when there are 0 customers coming to the drive-through) as well as the probability of a flurry of activity (when there are 5 or more customers coming to the drive-through). The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time from now until an earthquake occurs has an exponential distribution. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The formula is stated below:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^(-4) ] / 0!

= [ 1 * (2.7183)^(-4) ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^(-4) ] / 7!

= [ 16384 * (2.7183)^(-4) ] / 5040

= 0.05953877054

NAME: NWOKOLO DAVID OKECHUKWU

REG. NO: 2018/244291

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

A Poisson distribution can be used to analyze the probability of various events regarding how many customers go through the drive-through. It can allow one to calculate the probability of a lull in activity (when there are 0 customers coming to the drive-through) as well as the probability of a flurry of activity (when there are 5 or more customers coming to the drive-through). The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time from now until an earthquake occurs has an exponential distribution. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The formula is stated below:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^(-4) ] / 0!

= [ 1 * (2.7183)^(-4) ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^(-4) ] / 7!

= [ 16384 * (2.7183)^(-4) ] / 5040

= 0.05953877054

NAME: GINIKANDU IFECHUKWU JOSEPH-MARY

REG. NO: 2019/245716

DEPARTMENT: ECONOMICS

FACULTY: SOCIAL SCIENCES

1a) Poisson probability distribution:

A Poisson distribution can be used to analyze the probability of various events regarding how many customers go through the drive-through. It can allow one to calculate the probability of a lull in activity (when there are 0 customers coming to the drive-through) as well as the probability of a flurry of activity (when there are 5 or more customers coming to the drive-through). The formula is stated below:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^(-4) ] / 0!

= [ 1 * (2.7183)^(-4) ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^(-4) ] / 7!

= [ 16384 * (2.7183)^(-4) ] / 5040

= 0.05953877054

Anuonye Anomnachi Yabuikem

2019/246211

ECO 231

1a) Poisson probability distribution:

{ P(X=x) = (λ^x*e^-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ^0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe^(- λx) }

The mean = 1/ λ

The variance =1/ λ^2

The standard deviation =(1/ λ^2)^0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = p*q^x

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)^7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)^(-4) ] / 0!

= [ 1 * (2.7183)^(-4) ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λ^x*e^-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)^(-4) ] / 7!

= [ 16384 * (2.7183)^(-4) ] / 5040

= 0.05953877054

1a. POISSON PROBABILITY DISTRIBUTION: This is a probability distribution that is used to show how many times an event is likely to occur over a specified period. Poisson distributions are often used to understand independent events that occur at a constant rate, within a given interval of time.

b. UNIFORM PROBABILITY DISTRIBUTION: This is a type of probability distribution in which all outcomes are equally alike. A coin has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

c. EXPONENTIAL PROBABILITY DISTRIBUTION: This is the probability distribution of the time between events in a poisson point process, that is, a process in which events occur continuously and independently at a constant average rate.

2. N=26

n=7

r=6 doctors (because it appeared first)

x=4 doctors

P(X=x)=(r) (N-r)

(x) (n-x)

___________

(N)

(n)

P(X=x)=(6) (26-6)

(4) (7-4)

____________

(26)

(7)

= (6) (20)

(4) (3)

____________

(26)

(7)

= 6C4 * 20C3

______________

26C7

= 15 * 1140

____________

657800

= 17100

________

657800

= 0.026

x=7-3=4 (since nurses are failures)

P(X=4)= 0.026

Therefore, the probability that 4 doctors and 3

nurses are picked is:

0.026 * 0.026

= 0.000676

3. p=0.35 q=0.65

The actual x for geometric distribution is x-1,

which is 8-1=7

x=7

P(X=x)=pq^x

P(X=7)=(0.35) (0.65)^7

=0.017

4. P(X=x)=Lambda^x * e^-lambda

________________________

x!

Lambda (mean) = 50/12 = 4.17 = 4 (to the

nearest whole number)

Lambda = 4

a. x=0

P(X=x)=4^0 * e^-4

__________

0!

= 1 * 0.0183

____________

1

= 0.0183

b. x=7

P(X=7) = 4^7 * e^-4

___________

7!

= 16384 * 0.0183

_________________

5040

= 0.059

CHUKWUEMEKA FAVOUR CHIDUBEM

2019/242734

Chidoobem@gmail.com

1. Write short notes on the following types of probability distributions:

a. Poisson probability distribution: it is used when you’re interested in the number of times an event occurs in a given area of opportunity . An area of opportunity is an area in which more than one event can occur. In possion distribution, we have unlimited number of outcomes. Examples include:Number of mosquito bites on a person , number of car scratches on a car paint etc.this is very helpful for planning processes as it enables managers to analyze consumer’s behaviour.

b. Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

c. Exponential probability distribution: This is usually present, when we are dealing with events that are rapidly changing. For e.g news, blogsetc.It is the probability distribution of the time between events in a Poisson point process.

2. Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

!P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3.Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

4.To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

QUESTION ONE

POISSON PROBABILITY DISTRIBUTION: The probability distribution of a Poisson random variable X representing the number of successes occurring in a given time interval or a specified region of space is given by the formula

P(X=x)= lxel

x! where

{x}={0},{1},{2},{3}\ldotsx=0,1,2,3…

{e}={2.71828}e=2.71828 (but use your calculator’s e button

mu=μ= mean number of successes in the given time interval or region of space

The Poisson random variable satisfies the following conditions:

The number of successes in two disjoint time intervals is independent.

The probability of a success during a small time interval is proportional to the entire length of the time interval.

Apart from disjoint time intervals, the Poisson random variable also applies to disjoint regions of space.

The Poisson distribution is often used in situations where we are counting the number of successes in a particular region or interval of time, and there are a large number of trials, each with a small probability of success. For example, the following random variables could follow a distribution that is approximately Poisson.

• The number of emails you receive in an hour. There are a lot of people who could potentially email you in that hour, but it is unlikely that any specific person will actually email you in that hour. Alternatively, imagine subdividing the hour into milliseconds. There are 3.6×106 seconds in an hour, but in any specific millisecond it is unlikely that you will get an email.

• The number of chips in a chocolate chip cookie. Imagine subdividing the cookie into small cubes; the probability of getting a chocolate chip in a single cube is small, but the number of cubes is large.

• The number of earthquakes in a year in some region of the world. At any given time and location, the probability of an earthquake is small, but there are a large number of possible times and locations for earthquakes to occur over the course of the year.

The parameter λ is interpreted as the rate of occurrence of these rare events; in the examples above, λ could be 20 (emails per hour), 10 (chips per cookie), and 2 (earthquakes per year). The Poisson paradigm says that in applications similar to the ones above, we can approximate the distribution of the number of events that occur by a Poisson distribution.

b)UNIFORM PROBABILITY DISTRIBUTION: The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. A uniform distribution, also called a rectangular distribution, is a probability distribution that has constant probability.

This distribution is defined by two parameters, a and b:

a is the minimum.

b is the maximum.

The distribution is written as U(a, b).

When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. Let F be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that the inverse function F−1 exists, as a function from (0, 1) to R. We then have the following results.

1. Let U ∼ Unif(0, 1) and X = F−1 (U). Then X is an r.v. with CDF F. Let X be an r.v. with CDF F. Then F(X) ∼ Unif(0, 1).

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

C) EXPONENTIAL PROBABILITY DISTRIBUTION: The exponential distribution is a continuous probability distribution used to model the time elapsed before a given event occurs. SometimesSometimes it is also called negative exponential distribution. The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, marketing studies have shown that the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

QUESTION TWO

(P) = number of outcomes of A÷ total number of outcomes in S

Total number= 26. Probability of picking 4 doctors and 3 nurses is

4/26 + 3/26 = 7/26

QUESTION THREE

P(X=x)= Pq^x.

P= 0.35, q=1-p= 0.65, x= x-1= 8-1= 7

P(X=x) = (0.35)(0.65)^7

= 0.000003154

=3.154*10^-6

QUESTION FOUR

P(X=x)= e^-/\ * /\ ÷ x|

/\= 50÷12 = 4.1667= 4

P(X=0)= e^-4 * 4^0 /0|

= 0.0183

ii) P(X=7)= e^-4 * 4^7/7|

= 0.0595

1a.) Possion Distribution:

Poisson distribution is a type of discrete probability function that describes the number of times an event occurs in a specified period of time.It has to do with time and frequency.Its formula is=π^x–e^-π÷ x!. The π is the parameter of the distribution which indicates the number of events in a given time interval.The e in the distribution is permanent it is =2.7183.

B.)Uniform Distribution:

Uniform Distribution is a continuous distribution that assigns only positive probabilities within a specified interval.Uniform Distribution is a probability distribution where probability of x is constant.It is a form of probability distribution in which every possible outcome has equal chances of occurring.The probability is constant since each variable has equal opportunity of being the outcome.

C.)Exponential Distribution:

Exponential distribution is a continuous distribution that is commonly used to measure the expected time for an event to occur.It is defined as the reciprocal of the scale parameter and indicates how quickly decay of the exponential function occurs.Its formular is F(x;\π) = {πe^{-π X> 0 .0. X< 0}.

F(x;\π)=probability density function ;π=rate parameter; x=random variable.

2.)Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution:

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3.)Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

Solution:

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0172.

4.)To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda to the nearest whole number. What is the probability that it contains?

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value in a list Examples include number of mosquito bite on person, number of computer crashes in a day etc.

Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65⁷

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) 2.7183–⁵×5⁰/0

=0.0067×1/1

=0.0067

P(x=7) 2.7183-⁷×5⁷/5040

= 0.0067×78125/5040

=523.4/5040

=0.1038

INTERMEDIATE ECONOMICS STATISTICS (ECO 231)

a. Poisson probability Distribution: This is used when one is interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurance of an event can occur. Eg. Include; The number of mosquito bites in a person, The number of scratches in a cars paint. Etc.

Poisson Distribution formula is given as: ” Lambda sign raised to power of x times e(exponential) raised to power -lambda Sign. IE e -\ -\X

X!

b. Uniform probability distribution: Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely to occur. Now, the difference between Normal Distribution and Uniform Distribution is a probability distribution which peaks at the middle and gradually decreases. It is a probability distribution where probability of X is highest at the center and lowest in the ends WHEREAS Uniform probability distribution of X is constant.

The Distribution describes where there is an arbitrary outcome that lies between certain bounds.

The notation for Uniform distribution is X〰️U(a b) where a=lowest Value of X and b=highest Value of X.

c. Exponential Distribution: This is usually present when we are dealing with events that are rapidly changing.

In probability theory, the exponential distribution is defined as the probability of time between events in the Poisson point process. Also the exponential distribution is the continuous analogue of the geometric distribution.

2. Using the formula (xr) (n-xN-r )/(nN) Which is hypergeometric formula we solve thus

(46(320)/(726)= (6C4)( 20C3 )/ (26C7 ) =17100/657800=0.0260

3. P(1-P)x = 0.35( 1-0.35)7 = 0.35(0.049) = 0.01715

4. Using Poisson distribution. Where -\ = 50/12 =4.1~4

P(x=0) e -\ -\X = 2.7183-4 × 40 =0.0183

X! 0!

P(x=7) = 2.7183 -4 × 47 = 299. 8272 = 0.0595

7! 5040

Spencer Divine ezekwesiri

2019/243431

Economics.

chukwuebukah70@gmail.com

1a. POISSON PROBABILITY DISTRIBUTION.

A discrete frequency distribution which gives the probability of a number of independent events occurring in a fixed time. This is used when you are interested in the number of times an event occurs in a given area of operating. It is used in explaining time, frequency and rate.

B. UNIFORM PROBABILITY DISTRIBUTION.

In statistics, uniform probability distribution refers to a type of probability distribution in which all out comes are equally likely. It is a probability that asserts that the outcomes for a discrete set of data have the same probability.

C. EXPONENTIAL PROBABILITY DISTRIBUTION.

It is the probability distribution of the time between events in poison point process i.e a process in which events occurs continuously and independent at a constant average rate. This is usually present when we are dealing with events that are rapidly changing early on for. E.g news, blog.

2. P{X=doctors and 3 nurses}

N=6+20=26

n= 7, x=4 doctors, 3 nurses

P(X=x)=|rCx| [N-rCn-x]/NCn

=(6C4) (20C3)/26C7

=17100/657800= 0.026.

3. P(X=x) Pqx

P=0.35

Q=1-p=1-0.35=0.65

P(X=7)=0.35×0.66^7=0.0176

4a. P(X=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

4b. P(X=7) e^-50×50^7/7!

=0.0067×781.25/5040

= 523.4/5040

=0.1038.

Poisson probability distribution:This is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurance of an event can occur.

Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. For example; A coin has a uniform distribution because the probability of getting either head or tails in a coin toss is the same.

Exponential distribution is the probability distribution of the time between events in a poisson process i.e a process in which event occur continuously and independently at a constant average rate.

2. (r)(N-r)

(x)(n-x)

—————-

(N)

(n)

Where r=6

n=7

x=4

N=26

(6)(26-6). =(6C4)(20). =6C4×20C3

(4)(7-4). =(4) (3) ———————-

—————- —————-. 26C7

(26)

(7)

15×1140. = 17100

—————– —————-. =0.0260

657800 657800

3.Pqx

P=0.35. q=1-p

q=0.65

x=7

=0.35×0.65^7 =0.35×0.0490

=0.0172

4.π^x e^-x

When x=50/12 =4

e=2.7183

x=0

4^0×2.7183^-4 =1×0.0183=0.0183

————————. —————

0! 1

Ii. P(X=7)

When π=4, e=2.7183, x=7

4^7 ×2.7183^-4 = 16384×0.0183

————————- ————————

7! 5040

299.8272

—————–. = 0.0595

5040

Poisson probability distribution:This is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurance of an event can occur.

Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. For example; A coin has a uniform distribution because the probability of getting either head or tails in a coin toss is the same.

Exponential distribution is the probability distribution of the time between events in a poisson process i.e a process in which event occur continuously and independently at a constant average rate.

2. (r)(N-r)

(x)(n-x)

—————-

(N)

(n)

Where r=6

n=7

x=4

N=26

(6)(26-6). =(6C4)(20). =6C4×20C3

(4)(7-4). =(4) (3) ———————-

—————- —————-. 26C7

(26)

(7)

15×1140. = 17100

—————– —————-. =0.0260

657800 657800

3.Pqx

P=0.35. q=1-p

q=0.65

x=7

=0.35×0.65^7 =0.35×0.0490

=0.0172

4.π^x e^-x

When x=50/12 =4

e=2.7183

x=0

4^0×2.7183^-4 =1×0.0183=0.0183

————————. —————

0! 1

Ii. P(X=7)

When π=4, e=2.7183, x=7

4^7 ×2.7183^-4 = 16384×0.0183

————————- ————————

7! 5040

299.8272

—————–. = 0.0595

5040

Okoro Henry Chukwuebuka

Economics

2019/249001

chukwuebukah70@gmail.com

NAME: ANIEMEKA CHIJINDU DENNIS

REG. NO: 2019/250915

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: CHUKWUKAODINAKA JOHN OLUCHUKWU

REG. NO: 2019/245518

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

1a. Poisson probability distribution: This is used when you are interested in the number of times and event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time,volume,or such area in which more than one occurrence of an event can occur.

b. Uniform probability distribution: This is a type if probability distribution in which all possible outcomes are equally likely. It is divided into two types i.e discrete uniform distribution and continuous uniform distribution

c. Exponential probability distribution: This is a continuous probability distribution used to model the time elapsed before a given event occurs. It is often used to answer probabilistic terms questions.

2. Using hyper geometric distribution

P(X=x) = (rCx) (N-rCn-x)

________________

(NCn)

Where N= 26 i.e (6+20)

n = 7

r = 6

N – r = 20 (26 – 6)

n – x = 7 – 4 = 3

P(X= 4doctors & 3nurses) = (6C4) (20C3)

______________

(26C7)

=15 × 1140

____________ = 0.0253

657800

3. P(X=x) = P(1-p)^x or Pq^x

Let x be the number of relationships before the marriage

P(X=7) = 0.35 × (0.65)^7

0.35 × 0.049 = 0.0172

4. Landha = 50 ÷ 12 = 4.2 to the nearest whole number is 4

Using poisson distribution

P(X=x) = landha^x e^-landa

____________________

x!

a. P(X=0) = 4^0 e^-⁴

_________ = 0.0183

0!

b. P(X=7) = 4^7 e^-⁴

_________ = 299.8272 ÷ 5040

7! = 0.0595

NAME: OGUZIE ECHEZONACHUKWU SIXTUS

REG. NO: 2019/249165

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

Ikwuagwu Lucy Ogechi 2019/245407

1. a. Poisson probability distribution: this is used when interested in the number of times an event occurs in a given area of opportunity.

b. Uniform probability distribution: refers to a type of probability distribution in which all outcomes are equally likely.

c. Exponential probability distribution: Is often used to model the time elapsed between events.

2. rCx X (N-r)(n-x)/NCn , this is the formula for hypergeometric probability distribution

r=6, x=4, N=26, n=7

6C4*( 26 – 6 )C( 7 – 4 )/26C7

=15(1140)/657800

=0.026.

3. P( X=x) = Pq^x

= 0.35( 0.65 )^7

=0.0172

This is a geometric probability distribution

4. e^-∆ ∆^x/ x!= e^-50*50^0/0!

= 0.00000000000000000000010

ii. e^∆ ∆ ^x/x!= e^-50*50^7/7!= 0.000000000003

NAME; EZEH PATRICK EZENWA

REG NO-2019/244053

1. Write short notes on the following;

Poisson Distribution

In statistics, a Poisson distribution is a probability distribution that is used when you are interested in the number of times an event occurs in a given area of opportunity. In Poisson distribution there can be unlimited number of possibleoutcomes.it can also be said to show how many times an event is likely to occur over a specified period. In other words, it is called a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. The Poisson distribution is used in testing how unusual an event frequency is for a given interval. It was named after the French mathematician Siméon Denis Poisson. The Poisson distribution is also a discrete function, which means that the variable can only take specific values in an infinite list Its formula is: P(X=x)= e-ƛƛx/x!

Uniform Distribution

In statistics, uniform distribution is a type of probability distribution in which all outcomes are equally likely. A deck of cards for example has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails is equal or the same. Uniform probability has two types which are Discrete and Continuous. In a discrete uniform distribution, outcomes are discrete and have the same probability while in a continuous uniform distribution, outcomes are continuous and infinite. In a normal distribution, data around the mean occur more frequently and the frequency of occurrence decreases the farther you are from the mean in a normal distribution. The formula is written as: X~U(a,b).

Exponential Distribution

The exponential distribution is a continuous probability distribution that concerns the amount of time until some specific event happens. It is usually present when we are dealing with events that are rapidly changing early on for example, news, blog etc. It is a process in which events happen continuously and independently at a constant average rate. The exponential random variable is said to be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the market is an exponential distribution. Its formula is written as:

P(X=x)=me-mx=1µe-1µx, where, m= the parameter, and µ the average time of occurrence.

Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

NAME; EZEH PATRICK EZENWA

REG NO-2019/244053

1. Write short notes on the following;

a. Poisson Distribution

In statistics, a Poisson distribution is a probability distribution that is used when you are interested in the number of times an event occurs in a given area of opportunity. In Poisson distribution there can be unlimited number of possibleoutcomes.it can also be said to show how many times an event is likely to occur over a specified period. In other words, it is called a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. The Poisson distribution is used in testing how unusual an event frequency is for a given interval. It was named after the French mathematician Siméon Denis Poisson. The Poisson distribution is also a discrete function, which means that the variable can only take specific values in an infinite list Its formula is: P(X=x)= e-ƛƛx/x!

b. Uniform Distribution

In statistics, uniform distribution is a type of probability distribution in which all outcomes are equally likely. A deck of cards for example has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails is equal or the same. Uniform probability has two types which are Discrete and Continuous. In a discrete uniform distribution, outcomes are discrete and have the same probability while in a continuous uniform distribution, outcomes are continuous and infinite. In a normal distribution, data around the mean occur more frequently and the frequency of occurrence decreases the farther you are from the mean in a normal distribution. The formula is written as: X~U(a,b).

c. Exponential Distribution

The exponential distribution is a continuous probability distribution that concerns the amount of time until some specific event happens. It is usually present when we are dealing with events that are rapidly changing early on for example, news, blog etc. It is a process in which events happen continuously and independently at a constant average rate. The exponential random variable is said to be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the market is an exponential distribution. Its formula is written as:

P(X=x)=me-mx=1µe-1µx, where, m= the parameter, and µ the average time of occurrence.

2. Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

a. No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

b. 7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

CLEMENT ANN AMAKA

REG NO-2019/245757

1. Write short notes on the following;

POISSON DISTRIBUTION

In probability theory and statistics, the Poisson distribution, was named after a French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time

Poisson distribution, in statistics, a distribution function useful for characterizing events with very low probabilities of occurrence within some definite time or space. … Letting p represent the probability of a win on any given try, the mean, or average, number of wins (λ) in n tries will be given by λ = np

Poisson distribution is a discreet function meaning that the variable can only take a specific values in a list

Poisson distribution is a theoretical discrete probability and is also known as the Poisson distribution probability mass function. It is used to find the probability of an independent event that is occurring in a fixed interval of time and has a constant mean rate. The Poisson distribution probability mass function can also be used in other fixed intervals such as volume, area, distance, etc. A Poisson random variable will relatively describe a phenomenon if there are few successes over many trials. The Poisson distribution is used as a limiting case of the binomial distribution when the trials are large indefinitely. If a Poisson distribution models the same binomial phenomenon, λ is replaced by np

Poisson distribution is: f(x) = P(X=x) = (e-λ λx )/x!

Where:

x = 0, 1, 2, 3…

e is the Euler’s number

λ is an average rate of value and variance, also λ>0

UNIFORM DISTRIBUTION

uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

uniform distribution, in statistics, distribution function in which every possible result is equally likely; that is, the probability of each occurring is the same.

The theory of point sets and sequences having a uniform distribution. Uniform distribution theory is important in modeling and simulation, and especially in so-called Monte Carlo and quasi-Monte Carlo methods, and seeks to characterize and construct well distributed point sets and sequences.

Any situation in which every outcome in a sample space is equally likely will use a uniform distribution. One example of this in a discrete case is rolling a single standard die. There are a total of six sides of the die, and each side has the same probability of being rolled face up.

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. The probability density function is f(x)=1b−a f ( x ) = 1 b − a for a ≤ x ≤ n.

EXPONENTIAL DISTRIBUTION

Exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at aconstant average rate. It is a particular case of the gamma distribution.The Poisson distribution deals with the number of occurrences in a fixed period of time, and the exponential distribution deals with the time between occurrences of successive events as time flows by continuously. … The both distribution are used in queuing systems.

For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts.

The formula for the exponential distribution: P ( X = x ) = m e – m x = 1 μ e – 1 μ x P ( X = x ) = m e – m x = 1 μ e – 1 μ x Where m = the rate parameter, or μ = average time between occurrences.

2. Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

Name: Nduul Michael T.

Reg no: 2019\246514

1. Write short notes on:

A. Poisson Distribution

This was named after the French mathematician Siméon Denis Poisson. In Poisson distribution there can be unlimited number of possible outcomes because it measures events that occurs in a given area of opportunity. It can also be said to show how many times an event is likely to occur over a specified period. In other words, it is called a count distribution. The Poisson distribution is used in testing how unusual an event frequency is for a given interval. The Poisson distribution is also a discrete function, which means that the variable can only take specific values in an infinite list Its formula is: P(X=x)= e-ƛƛx/x!. Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

B. Uniform Distribution

The uniform distribution is a type of probability distribution in which all outcomes are equally likely. A deck of cards for example has a uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely likewise is flipping a coin, you either get a head or a tail. Uniform probability has two types which are Discrete and Continuous. In a discrete uniform distribution, outcomes are discrete and have the same probability while in a continuous uniform distribution, outcomes are continuous and infinite. In a normal distribution, data around the mean occur more frequently and the frequency of occurrence decreases the farther you are from the mean in a normal distribution. The formula is denoted as: X~U(a,b).

C. Exponential Distribution

Exponential distribution is a continuous probability distribution that concerns the amount of time until some specific event happens. It is a process in which events happen continuously and independently at a constant average rate. The exponential random variable can be either more small values or fewer larger variables. For example, the amount of money spent by an individual on one trip to the mall is an exponential distribution. Its formula is written as:

P(X=x)=me-mx=1µe-1µx, where, m= the parameter, and µ the average time of occurrence.

2. Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

COURSE CODE: ECO 231

NAME: UMEH OGECHUKWU MARYANN

REG NO: 2019/243546

1a) In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

b) In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less.

2)

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

COURSE CODE: ECO 231

NAME: Ubazoro Chukwumeka George

REG NO: 2019/251195

1a) In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

b) In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less.

2)

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

COURSE CODE: ECO 231

NAME: Abonyi Kosiso Sunday

REG NO: 2019/244009

1a) In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

b) In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less.

2)

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

NAME: ONWUEGBUNA PRECIOUS ONYINYE

REG. NO: 2019/245507

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

1. Poisson probability distribution:

In probability theory and statistics, the Poisson distribution named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

The Poisson Distribution formula is: P(x; μ) = (e^-μ) (μ^x) / x!

Uniform Distribution:

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

Exponential distribution:

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

The formula for the exponential distribution: P ( X = x ) = m e – m x = 1 μ e – 1 μ x P ( X = x ) = m e – m x = 1 μ e – 1 μ x Where m = the rate parameter, or μ = average time between occurrences.

2. P(x-r) =. (rCx * N-rCn-x)/ NCn

N= 6+20 =26

r=6

n=7

X =4

P(X=4)= (6C4 × 26-6C7-4)\ 26C7

P(X=4) = (6C4 × 20C3)/26C7

P(X=4)= 0.0024995743

~0.0025

3. P(X=X) = owe

P= 0.35

q= 1-0.35 = 0.65

P(X=7)= 0.35×0.65^7

0.35×0.0490

= 0.0172

4. P(x; μ) = (e^-μ) (μ^x) / x!

e= 2.7183

μ= 50/12 = 4.167

I. P(X=0) =

(2.7183^-4.167)(4.167^0)/0!

P(X=0) = 0.015498

~0.0155

ii. P(X=7) =

(2.7183^-4.167)(4.167^7)/7!

(0.015498×21815.4664)/5040

=0.06708

~0.067

NAME: Chidobelu Yonna Raluchukwu

REG. NO: 2019/244261

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

COURSE CODE: ECO 2311a) In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

b) In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less.

NAME: Ezeamama Ifechukwu Emmanuel

REG NO: 2019/245102

2)

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

COURSE CODE: ECO 2311a) In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

b) In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less.

NAME: Nebechi Chinedu Joshua

REG NO: 2019/250115

2)

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

NAME: OKE AMARACHUKWU NNENNA

REG. NO: 2019/241949

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The probability density function is given as;

{ P(X=x) = (λxe-λ) / x! }

where, e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of x when that is also equal to its variance

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. The probability density function is given as:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

The mean = 0

The variance = 1

The standard deviation = 1

c) Exponential probability distribution:

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes, it is found in various other contexts. The probability density function is given as;

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

Name: NAME:- ONWUKWE JOSEPH NWACHUKWU2. Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

REG NO:- 2019/243773

1. Write short notes on:

A. POISSON PROBABILITY DISTRIBUTION

In probability theory and statistics, the Poisson distribution (/ˈpwɑːsɒn/; French pronunciation: [pwasɔ̃]), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.[1] The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

B. UNIFORM PROBABILITY DISTRIBUTION

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

C. EXPONENTIAL PROBABILITY DISTRIBUTION

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

Solution3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

1a) POISSON PROBABILITY DISTRIBUTION

A Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list.

Formular:

P(X:x) = (e^-λ λ^x )/x!

Where:

e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of x when that is also equal to its variance.

1b) UNIFORM PROBABILITY DISTRIBUTION

A uniform distribution, also called a rectangular distribution, is a probability distribution that has constant probability.

This distribution is defined by two parameters, a and b:

a is the minimum.

b is the maximum.

The distribution is written as U(a, b). Like all probability distributions for continuous random variables.

1c) EXPONENTIAL PROBABILITY DISTRIBUTION

Exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

The exponential distribution has the memoryless property, which says that future probabilities do not depend on any past information.

2) P(X:x) =(rCx) (N-rCn-x)/NCn

N = 6+20 = 26

n = 7

r = 6

x = 4

P(X:x) = (6C4)(26-6C7-4)/(26C7)

(6C4)(20C3)/(26C7)

15 * 1140/657800 = 0.026.

3) P(X:x) = PQ^x

P = 0.35

Q= 1-0.35= 0.65

x = 8-1=7

P(X=7) = (0.35)(0.65)^7 = 0.0172

The is approximately a 1.7% chance of her getting married in her 8th relationship.

4) P(X:x) = (e^-λ λ^x )/x!

e = 2.71828

λ = 50/12 = 4 to the nearest whole number.

a) x = 0

P(X=0) = (e^-4)(4^0)/0!

= 0.01832.

b) x = 7

P(X=7) = (e^-4)(4^7)/7!

= 0.0595

1a. Poisson probability distribution: In statistics, a poisson probability distribution is a distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician. The poisson distribution is a discrete function and its variables take 0,1,2,3, etc.

b. Uniform probability distribution: This is a distribution in which every possible result is equally likely, that is, the probability of each occuring is the same. It is the simplest form of distribution and sometimes used as null hypothesis or initial hypothesis.

c. Exponential probability distribution: This is one of the widely used continuous distributions. It is often used to model the time elapsed between events. It is commonly used to measure the expected time for an event to occur.

2. P(X=x)=(r) (N-r)

(x) (n-x)

____________

(N)

(n)

i. N=26

n=7

r=6

x=4

(6) (26-6)=(6) (20)

(4) (7-4) (4) (3)

_________ _________

(26) (26)

(7) (7)

= 6C4 × 20C3

____________

26C7

= 0.02599

Ii. P(X=3)=(6) (26-6)=(6) (20)

(3) (7-3)= (3) (4)

__________ _________

(26) (26)

(7) (7)

= 6C3 × 20C4

______________

26C7

= 0.14730

3. P(X=x)=p(1-p)^x-1 or pq^x-1

p=0.35, q=1-0.35=0.65

=0.35 × 0.65^7

=0.01715

4. P(X=x)=(λ^x × e^-λ)/x!

λ=50/12=4.17=4 (to the nearest whole number), e=2.7183, x=0

i. P(X=0)=4^0 × 2.7183^-4

__________________

0!

=0.018315

ii. P(X=7)=4^7 × 2.7183^-4

_________________

7!

=16384 × 0.18315

__________________

5040

= 0.5953

NAME: OFFOR UGOCHUKWU IKENNA

REG NO: 2019/245050

E-MAIL: ugosagacious@gmail.com

1. The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specific time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools.

2. The Exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

3. The Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely.

4. r = 6, N= 26, n = 7,

=

=

= 6C4 * 20C3/ 26C7 = 0.0259 Ans.

5. Pq^x-1 or P(1-P)^x-1; where P= 0.35, q= 1-0.35 = 0.65, x-1 = 8-1 = 7

P(X=7) = 0.35 * (0.65)^7 = 0.35 * 0.0490 = 0.0172 Ans.

6. P(X=x) = e-^ * ^x / x! P(X= 0)

= e-50 * 50^0 / 0! = 1.9287 *1/1 = 1.9287 Ans.

I. P(X= 7) = e-50 * 50^7 / 7! = 1.9278* 7.8125 / 5040 = 298,967,633.91 Ans.

1.

A: Poisson probability distribution is used to shows how many times an event is likely to occur over a specified period.

Poisson distribution is a discrete function meaning that the variable can only take specific values in a list.

B: Uniform Distribution:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

The probability density function is f(x)=1b−a(x)=1b−a for a ≤ x ≤ b.

C; Exponential probability distribution;

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.

Number Two ;

A; N=26,n=7 ,r=6, x=4

6C4x20C3 /26C7

=17100 /657,800

=0.026

B; 3 nurses

P(x=x)

=7-3 ;x=4

P(X=4)

6C4x20C3 /26C7

=15X1140 /657800

=0.026

0.026X100

=2.6%

FRANCIS CHINEDU MICHAEL

2019/244161

michael23fc@gmail.com

1i. Poisson distributions, valid only for integers on the horizontal axis. λ (also written as μ) is the expected number of event occurrences.With the Poisson distribution, companies can adjust supply to demand in order to keep their business earning good profit. In addition, waste of resources is prevented.

1ii. uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

1iii. exponential distribution is often concerned with the amount of time until some specific event occurs.examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. Exponential distribution is widely used in the field of reliability.

2. Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

!P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3.Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

3.To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

Number Three;

P=0.35

Q=1-p

q=1-0.35=0.65

X=8-1, x=7

P(X=7)= (0.35)(0.65)^7

=0.35 X 0.049 =0.0172

0.0172 X100 =1.7chances

FRANCIS CHINEDU MICHAEL

2019/244161

michael23fc@gmail.com

1i. Poisson distributions is valid only for integers on the horizontal axis. λ (also written as μ) is the expected number of event occurrences.With the Poisson distribution, companies can adjust supply to demand in order to keep their business earning good profit. In addition, waste of resources is prevented.

1ii. uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

1iii. exponential distribution is often concerned with the amount of time until some specific event occurs.examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. Exponential distribution is widely used in the field of reliability.

!P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3.Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

3.To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

EZE DANIEL UCHENNA

2018/244280

Ezedaniel021@gmail.com

1. Write short notes on the following types of probability distributions:

a. Poisson probability distribution: it is used when you’re interested in the number of times an event occurs in a given area of opportunity . An area of opportunity is an area in which more than one event can occur. In possion distribution, we have unlimited number of outcomes. Examples include: number of marker strokes on the white board, number of car scratches on a car paint etc

b. Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

c. Exponential probability distribution: This is usually present, when we are dealing with events that are rapidly changing. For e.g. news, blogs etc. It is the probability distribution of the time between events in a Poisson point process. exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

3.To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

Number Four;

A; P(X=0)

=50^0 X 2.7183^-50 /1

=5.185

=5

B; 50^7 X 2.7183^-50 /7!

=7.8125 X 5.185 /5,040

=0.00804

=0.00804 X 100

=0.804

= 1

1) Poisson Probability Distribution.

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

2) Uniform Probability Distribution

uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

3) Exponential probability Distribution

exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

2

N= 26 (6+20)

n = 7

x = 4

r = 6

i) Probability of picking 4 doctors

P(X=4) = 6 C 4 * 26-6 C 7-4 / 26 C 7

= 6 C 4 * 20 C 3 / 26 C 7

Therefore, P(X = 4) = 0.0259

Without replacement

ii) Probability of picking 3 nurses

P(X = 3)

x = 3

N = 26

r = 6

n = 3

P(X = 3) = 6 C 3 * 26-6 C 3-3 / 26 C 3

= 6 C 3 * 20 C 0 / 26 C 3

Therefore, P(X = 3) = 0.007692

3

Using geometric approach

P(X = x) = Pq^x

Where;

P = 0.35

q = 0.65 ( i.e, 1-0.35)

x = 7 (i.e, 8-7)

P(X = 7) = 0.35 * 0.65^7

= 0.01716

4

Using Poisson Probability Distribution

P(X = x) = Lambda^x * e^-lambda / x!

Where; lambda is the average mean

e = 2.7183 ( constant )

Lambda = 50/12 = 4.1667

i) Probability that it contains no chocolate chips

P(X = 0) = 4.1667^0 * 2.7183^-4.1667 / 0!

P(X = 0) = 0.01550

ii) probability that it contains 7 chocolate chips

P(X = 7) = 4.1667^7 * 2.7183^-4.1667 / 7!

P(X = 7) = 0.0000671

1) Poisson Probability Distribution.

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

2) Uniform Probability Distribution

uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

3) Exponential probability Distribution

exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

2

N= 26 (6+20)

n = 7

x = 4

r = 6

i) Probability of picking 4 doctors

P(X=4) = 6 C 4 * 26-6 C 7-4 / 26 C 7

= 6 C 4 * 20 C 3 / 26 C 7

Therefore, P(X = 4) = 0.0259

Without replacement

ii) Probability of picking 3 nurses

P(X = 3)

x = 3

N = 26

r = 6

n = 3

P(X = 3) = 6 C 3 * 26-6 C 3-3 / 26 C 3

= 6 C 3 * 20 C 0 / 26 C 3

Therefore, P(X = 3) = 0.007692

3

Using geometric approach

P(X = x) = Pq^x

Where;

P = 0.35

q = 0.65 ( i.e, 1-0.35)

x = 7 (i.e, 8-7)

P(X = 7) = 0.35 * 0.65^7

= 0.01716

4

Using Poisson Probability Distribution

P(X = x) = Lambda^x * e^-lambda / x!

Where; lambda is the average mean

e = 2.7183 ( constant )

Lambda = 50/12 = 4.1667

i) Probability that it contains no chocolate chips

P(X = 0) = 4.1667^0 * 2.7183^-4.1667 / 0!

P(X = 0) = 0.01550

ii) probability that it contains 7 chocolate chips

P(X = 7) = 4.1667^7 * 2.7183^-4.1667 / 7!

P(X = 7) = 0.0000671

1.

a. Poisson distribution is a type of discrete probability distribution that is used when interested in the number of times an event occurs in a given area of opportunity. Here, there is an unlimited number of possible outcomes. Some examples are the number of car accidents on a particular road, number of injuries on a person’s body.

b. Uniform probability distribution is a distribution in which possible outcomes are equally likely to occur. For example, tossing a coin. The chances of getting a head or tail are equal.

c. Exponential probability distribution enables individuals (or groups of persons) to measure the expected time for an event to occur. It is usually present when dealing with events that change frequently. For example, post or updates on a blog.

2.

N= 6 doctors + 20 Nurses =26

n= 4 doctors + 3 nurses = 7

r= 6, x= 4, N= 26, n= 7

Using the Hyper Geometric distribution;

P(X=x)

(rCx× N-rCn-x) ÷ NCn

(6C4 × 20C3) ÷ 26C7

(15 × 1140) ÷ 657800

= 0.0259

3.

p=0.35

q= (1- p)

(1- 0.35) = 0.65

x= 8

P(X=x) = pq^x

=> 0.35× 0.65^8

=> 0.35× 0.03186

=> 0.011

4.

n= 12 cookies

In a batch of 12 cookies = 50 chocolate chips

1 cookie = (50 ÷ 12) = 4.167

~ 4 chips in 1 cookie

Therefore,

h=4

P(X=x) (e^(-h)×h^x) ÷ x!

a. When x=0

=> (e^(-4)× h^0) ÷ 0!

=> 0.01832

b. When x=7

=> (e^(-4)× h^7 ) ÷ 7!

(0.01832 ×16,384) ÷ 5,040

=> 0.0595

~ 0.06

1a. POISSON PROBABILITY DISTRIBUTION

Poisson distribution often appears in connection with the study of sequences of random events occurring over time or space. Suppose, starting from a time point t = 0, we start counting the number of events. Then for each value of t, we obtain an integer denoted by N (t), which is the number of events that have occurred during the time period [0, t].

1b. UNIFORM PROBABILITY DISTRIBUTION

uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely.

1c. EXPONENTIAL PROBABILITY DISTRIBUTION

Exponential Probability Distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution

2. 4/26 × 3/25 = 12/650 (since it is without replacement).

3. Geometric = pq^x

= 0.35 × 0.65⁷

= 0.017

(p is 0.35, q = 1-0.35 = 0.65, No of failures = 7).

4. ¥ × e^-¥/x!

Where ¥ represents lambda

a. Where ¥ = 50

x = 0

Poisson distribution = 50⁰ × e ‐⁵⁰ ÷ 0!

= 1.92875 × 10^‐²²

b. Where ¥ = 50

x = 7

Poisson distribution = 50 ⁷× e ‐⁵⁰ ÷ 7!

= 2.98975 × 10^‐¹⁴

NAME: OMEYE ADANNA NGOZIKA

REG NO: 2019/242941

DEPT: ECONOMICS

1A) The Poisson Distribution is a tool used in probability theory statistics to predict the amount of variation from a known average rate of occurrence, within a given time frame.

B) In statistics, uniform distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. The probability is constant since each variable has equal chances of being the outcome.

C)The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events.

2) (6C4 × 20C3) ÷ 26C7

= (15×140)÷657800

= 0.0259

3) p=0.35

x=8

q=0.65

PQ^x= 0.35×0.65^8 = 0.011

4) In 1 batch of 12 there are 50 chips

To find the average = 50÷12= 4.16

LAMBDA= 4

i) P(X=0) (e^- lambda × lambda^x) ÷ x!

(e^-4 ×4^0) ÷ 1! = 0.01832

ii) P(X=7) (e^-4 × 4^7)÷7!

= (0.01832×16384)÷ 5040

= 0.0595

= 0.06

UGOCHUKWU GOODNESS ANULIKA

2019/244160

goodness.ugochukwu.244160@unn.edu.ng

Poisson Probability Distribution

The Poisson Distribution is a tool used in probability theory statistics to predict the amount of variation from a known average rate of occurrence, within a given time frame.In other words, if the average rate at which a specific event happens within a specified time frame is known or can be determined (e.g., Event “A” happens, on average, “x” times per hour), then the Poisson Distribution can be used as follows:

To determine how much variation there will likely be from that average number of occurrences

To determine the probable maximum and minimum number of times the event will occur within the specified time frame.

Poisson Distribution theme

Companies can utilize the Poisson Distribution to examine how they may be able to take steps to improve their operational efficiency. For instance, an analysis done with the Poisson Distribution might reveal how a company can arrange staffing in order to be able to better handle peak periods for customer service calls.

Uniform Probability Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50

Exponential Probability Distribution

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

2019/245070

Okafor chukwubuikem Emmanuel

Okarforchukwubuikem1@gmail.com

1a) Poisson Distribution

Poisson distribution, in statistic, a distribution function useful for characterizing events with very low probabilities of occurrence within some definite time or space. The French mathematician Simeon-Denis poison developed his function in 1830 to describe the number of times a gambler would win a rarely won game of chance in a large number of tries. Letting p represent the probability of a win on any given try, the mean, or average, number of wins (λ) in n tries will be given by λ = np. Using the Swiss mathematician Jakob Bernoulli’s binomial distribution, Poisson showed that the probability of obtaining k wins is approximately λk/e−λk!, where e is the exponential function and k! = k(k − 1)(k − 2)⋯2∙1. Noteworthy is the fact that λ equals both the mean and variance (a measure of the dispersal of data away from the dispersal of data away from mean) from the Poisson distribution

b) The Uniform Probability Distribution

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

c) The Exponential Probability Distribution

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. We will now mathematically define the exponential distribution, and derive its mean and expected value. Then we will develop the intuition for the distribution and discuss several interesting properties that it

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

OBASI SARAH CHINONSO

2019/250357

obasisarah001@gmail.com

1. Write short notes on the following types of probability distributions:

a. Poisson probability distribution:

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. Iit is used when you want to know number of times an event occurs in a given area of opportunity .In possion distribution, we have unlimited number of outcomes. Examples include:Number of mosquito bites on a person , number of car scratches on a car paint etc.s, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

In other words, A Poisson distribution, named after French mathematician Siméon Denis Poisson, can be used to estimate how many times an event is likely to occur within “X” periods of time

b. Uniform probability distribution:. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. it is also known as Continuous uniform distribution or rectangular distribution

.c. Exponential probability distribution: This is present, when we are dealing with events that are rapidly changing. For e.g news, blogsetc.It is the probability distribution of the time between events in a Poisson point process.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

Name:Onyisi sunny Hope

Reg no:2019/251206

Email:angelicprincess@gmail.com

1. Write short notes on the following types of probability distributions:

a. Poisson probability distribution:?

Ans:

A Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

b. Uniform probability:

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

c. Exponential probability distribution:

Exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

4.To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

2019/245394

Ngwoke Chidera Lilian

1. a. Poisson probability distribution: this is used when interested in the number of times an event occurs in a given area of opportunity.

b. Uniform probability distribution: refers to a type of probability distribution in which all outcomes are equally likely.

c. Exponential probability distribution: Is often used to model the time elapsed between events.

2. rCx X (N-r)(n-x)/NCn ,

r=6, x=4, N=26, n=7

6C4*( 26 – 6 )C( 7 – 4 )/26C7

=15(1140)/657800

=0.026.

3. P( X=x) = Pq^x

= 0.35( 0.65 )^7

=0.0172

This is a hypergeometric Probability distribution

4. This is a geometric probability distribution

e^-∆ ∆^x/ x!= e^-50*50^0/0!

= 0.00000000000000000000010

ii. e^∆ ∆ ^x/x!= e^-50*50^7/7!= 0.000000000003

1a.) A Poisson probability distribution, According to Britannica ,is a distribution function useful for characterizing events with very low probabilities of occurrence within some definite time or space. The French mathematician Siméon-Denis Poisson developed his function in 1830 to describe the number of times a gambler would win a rarely won game of chance in a large number of tries. Letting p represent the probability of a win on any given try, the mean, or average, number of wins (λ) in n tries will be given by λ = np. Using the Swiss mathematician Jakob Bernoulli’s binomial distribution, Poisson showed that the probability of obtaining k wins is approximately

P(k)=( λ^k )(e^(-λ) )/k!

where e is the exponential function and k! = k(k − 1)(k − 2)⋯∙1. Noteworthy is the fact that λ equals both the mean and variance (a measure of the dispersal of data away from the mean) for the Poisson distribution.

Thus the Poisson distribution is derived from the Bernoulli distribution by introducing λ and finding the limit with n going to infinity.

1b.)A uniform probability distribution, also called a rectangular distribution, is a probability distribution that has constant probability. It can be continuous or discrete and can be visualized as a vertical line for instance the probability of getting heads or tails is 0.5 each , it is uniform as it has constant probability for all outcomes.

1c.)The exponential probability distribution is often concerned with the amount of time until some specific event occurs. It is a continuous probability distribution since it deals with time and it is largely applied in businesses. In exponential distributions there are smaller number of large variables and higher number of small variables. For example for a business man there are small number of cases where transactions involve millions and many cases where it involves thousands. The probability density function is given by:

p(x)=(1/μ) e^(-1/μ x)

Where μ is the average . And has a mean and standard deviation of 1/μ.

2.) N=population=26

r= number of doctors which is success in this case = 6

N-r=number of nurses which here is considered failure = 20

n=sample =7 , x=success in sample=4, n-x=failure in sample=3

Using formula for hypergeometric probability distribution ( which is what we use here because it is without replacement )

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.) Using formulary for geometric distribution( since it concerns the number of failures before success)

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)using Poisson distribution (since we are concerned with time and averages )

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4 (to the nearest whole number)

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

(Since anything raised to the power of zero and 0! Factorial is 1)

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

NAME: OJOMAH FAVOUR ONYEKACHUKWU

REG NUMBER: 2019/244245.

DEPARTMENT: ECONOMICS

EMAIL: Ojomahfavour2@gmail.com

1a. Poisson Probability Distribution: This is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here, there can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes i.e success or failure.

Examples include: The number of car accidents over a certain range of a road, the number of computer crashes in a day.

1b. Uniform Probability Distribution: In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade.

1c. Exponential Probability Distribution: This is usually present when we are dealing with events that are rapidly changing early on for example news, blog.

2. Six doctors and twenty nurses attend a small Conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 Doctors and 3 Nurses are picked ?

Solution

N = 26 ( 6 + 20 =26)

n = 5, r = 6, x = ?

6C4X20C3/26C7

=17100/657800

= 0.0260.

3. Using the geometric, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

= P(1-p)^x or pq^x

P = 0.35 1- P = 0.65 x = 7

= 0.35(0.65)^7

0.35 X 0.0490 = 0.0172

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. if a cookie is chosen at random from a large batch, also using lambda (the mean) to the nearest whole number. What is the probability that it contains:

a, No chocolate chips?

b, 7 chocolate chips?

Solution

a. Mean = to the nearest whole number 4

Mean = 4, x = 0

P(X=0) = 2.7183^-4 (4)^0/0!

=0.0183X1

= 0.0183

B, 7 Chocolate Chips?

P(X=7) = 2.7183^-4 (4)^7/7!

= 0.0183×16384/5040

= 299.8272

=0.059

1a.) A Poisson probability distribution, According to Britannica ,is a distribution function useful for characterizing events with very low probabilities of occurrence within some definite time or space. The French mathematician Siméon-Denis Poisson developed his function in 1830 to describe the number of times a gambler would win a rarely won game of chance in a large number of tries. Letting p represent the probability of a win on any given try, the mean, or average, number of wins (λ) in n tries will be given by λ = np. Using the Swiss mathematician Jakob Bernoulli’s binomial distribution, Poisson showed that the probability of obtaining k wins is approximately

P(k)=( λ^k )(e^(-λ) )/k!

where e is the exponential function and k! = k(k − 1)(k − 2)⋯∙1. Noteworthy is the fact that λ equals both the mean and variance (a measure of the dispersal of data away from the mean) for the Poisson distribution.

Thus the Poisson distribution is derived from the Bernoulli distribution by introducing λ and finding the limit with n going to infinity.

1b.)A uniform probability distribution, also called a rectangular distribution, is a probability distribution that has constant probability. It can be continuous or discrete and can be visualized as a horizontal line for instance the probability of getting heads or tails is 0.5 each , it is uniform as it has constant probability for all outcomes.

1c.)The exponential probability distribution is often concerned with the amount of time until some specific event occurs. It is a continuous probability distribution since it deals with time and it is largely applied in businesses. In exponential distributions there are smaller number of large variables and higher number of small variables. For example for a business man there are small number of cases where transactions involve millions and many cases where it involves thousands. The probability density function is given by:

p(x)=(1/μ) e^(-1/μ x)

Where μ is the average . And has a mean and standard deviation of 1/μ.

2.) N=population=26

r= number of doctors which is success in this case = 6

N-r=number of nurses which here is considered failure = 20

n=sample =7 , x=success in sample=4, n-x=failure in sample=3

Using formula for hypergeometric probability distribution ( which is what we use here because it is without replacement )

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.) Using formulary for geometric distribution( since it concerns the number of failures before success)

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)using Poisson distribution (since we are concerned with time and averages )

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4 (to the nearest whole number)

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

(Since anything raised to the power of zero and 0! Factorial is 1)

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

1a. POISSON PROBABILITY DISTRIBUTION: This is a discrete random variable. It is a probability distribution that is used to show how many times an event is likely to occur over a specified period. The variable of a poisson distribution can only take specific values in a list.

1b. UNIFORM PROBABILITY DISTRIBUTION: It refers to a type of probability distribution in which all outcomes are equally likely. In a discrete uniform distribution, outcomes are discrete and have the same probability while in a continuous uniform probability outcome are continuous and infinite.

1c. EXPONENTIAL PROBABILITY DISTRIBUTION: It is one of the popularly or widely used continuous distributions, it is often used to model the time elapsed between events.

2. rCx × (N-r)C(n-x)÷NCn

Where r=6

n=7

N=26

x=4

= 6C4 × (26-6)C(7-4)÷26C7

= 6C4 × 20C3÷26C7

=15×1140÷657800

=0.0260

3. pq^x

Where p=0.25

q=1-p =1-0.25 =0.65

x=7

= 0.25×0.65^7

= 0.25×0.0490

= 0.0172

4. P(X=x) = √^x × £^-2

Where √= lander=50/12 =4.17

£= exponatial

4a. Where x=0

P(X=x) =

4.17^0 × £^-4.17/0!

1 × 0.155/1

P(X=0) = 0.155

4b. Where x=7

P(X=7) =

4.17^7 × £^-4.17/7!

=21925.65 ×0.015/5040

= 338.80/5040

P(X=7)= 0.0672

Name: Abasilim Chisom Judith

Reg no: 2019/249128

Email: abasilimchisom@gmail.com

1.Poisson probability distribution: A Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specific period of time. Poisson distribution is often used to understand independent events that occur at a constant rate within a given interval of time. A Poisson distribution is a discrete function, it means that the variable can only take specific values in a (potentially infinite) list, the variables can only take the values 0,1,2,3,4 and so on… without any fractions or decimals. Formula=P(X=x)=(λ^x × e^-λ)/x!

b. Uniform probability distribution: This refers to a type of distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. Formula= P(X=x)=1/b-a

There are two types of uniform probability distribution and they are; discrete and continuous. The possible result for rolling a die provide an example of discrete uniform distribution, it is possible to roll a 1,2,3,4,5 or 6 but it is not possible to roll a 2.3,4.5 or 5.3. some uniform distribution is continuous rather than discrete. An idealized random number generator would be considered a continuous uniform distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there’s an infinite number of points between 0.0 and 1.0.

C. Exponential probability distribution: The exponential distribution is the probability distribution of the time between events in a Poisson point process that is a process in which events occur continuously and independently at a constant average rate. In other words, an exponential probability distribution is a continuous distribution that is commonly used to measure the expected time of an event to occur. Formula=P(X=x)= λe^-λx

2.

P (4 docs and 3 nurses):

r=6

n=7

N=26

X= 4 docs and 3 nurses

P(X=4)= [rCx] . [N-rCn-x]/NCn

= (6C4)(20C3)/26C7

=15 . 1140/657800

=17100/657800

=0.026.

Then for nurses x= n-x = 7-3=4

(6C4)(20C3)/26C7

=0.026

Therefore p( 4doctors and 3 nurses) = 0.026 . 0.026

= 0.0076

3. Formula for geometric distribution; pq^x

P=0.35, x=8-1 = 7, q= 1- 0.35 = 0.65

0.35(0.65)

P(x=7) = 0.35 × 0.65^7

= 0.0176.

4. To make a dozen chocolate chip cookies, 50 chocolate chips are mixed into the dough, the same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lambda to the nearest whole number, what is the probability that it contains?

a. No chocolate

b. 7 chocolate

Lambda = 50/12 = 4.16

Lambda = 4( to the nearest whole number)

a. P(x=0) =2 . 718^ -4 . 4^0/0!

=0.0183 . 1/1

=0.0183

b. P(x=7) = 2.718^-4 . 4^7 /7!

=0.0183 . 16384/5040

=299.8/5040

=0.0594

1a)POISSON PROBABILITY DISTRIBUTION

This is a discrete random variables that is often useful in estimating the number of occurrences over a specified interval of time or space. For example, the random variable of interest might be the number of arrivals at a car wash in one hour, the number of repairs needed in 10 miles of highway or the number of leaks in 100 miles of pipeline. In Poisson probability distribution, the probability of an occurrence is the same for any two intervals of equal length. Also, the occurrence or non-occurrence in any other interval is independent of the occurrence in any other interval. The Poisson probability distribution function is

P(x)=e^-√•√*/x!

Where u is the distribution mean and can also be represented by lambda and x is the number of expected.

1b) UNIFORM PROBABILITY DISTRIBUTION

This is a continuous discrete random variables that its probability is always proportional to the length of the interval. It shows that a random variable value within a specified interval.For example,a uniform probability distribution question can be what is probability that the flight time is between 120 and 130 minutes?

P(x)={1/b-a} or 1b-a for a<x

0, u>0 where x is the number of time and u is the average number of times an event occurred.2)nCr=n!/r!(n-r)! We solve for the problem by using combination because it has to do with selection.

Step 1: solve for the total number of Doctors selected

n=6,r=4

nCr =n!/r!(n-r)! = 6!/4!(6-4)!=15

Step 2:Solve for the total number of Nurses selected

n=20,r=3

nCr=n!/r!(n-r)!=20!/3!(20-3)!=1140

The total number of people selected =the total number of Doctors selected + the total number of Nurses selected =15+1140 =1155

3) Geometric probability distribution function = pq* or p(1-q)*

P= 0.35

q=1-p=1-0.35=0.65

X=8-1=7

P(x)=pq*=0.35(0.65)^7 =0.35(0.4902)=0.0172

4)This is a poisson probability distribution function is

P(x)=e^-√•√

√(mean)=50 chocolate cookies/12 dozens of chocolate cookies=4.16~4

A)No chocolate chips

√=4,x=0

P(0)=e^-4•4^0/0! =0.0183

P(x)=0.0183

B)7 chocolate chips

X=7,√=4

P(7)=e^-4•4^7/7!

P(7) =0.0183•16384/5040=0.0595

P(X)=0.0595

Odo Linda Amarachi

2019/244376

lindaammy162@gmail.com

1. Poisson probability distribution:

In probability theory and statistics, the Poisson distribution named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

The Poisson Distribution formula is: P(x; μ) = (e^-μ) (μ^x) / x!

Uniform Distribution:In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

Exponential distribution:

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

The formula for the exponential distribution: P ( X = x ) = m e – m x = 1 μ e – 1 μ x P ( X = x ) = m e – m x = 1 μ e – 1 μ x Where m = the rate parameter, or μ = average time between occurrences.

2. P(x-r) =. (rCx * N-rCn-x)/ NCn

N= 6+20 =26

r=6

n=7

X =4

P(X=4)= (6C4 × 26-6C7-4)\ 26C7

P(X=4) = (6C4 × 20C3)/26C7

P(X=4)= 0.0024995743

~0.0025

3. P(X=X) = owe

P= 0.35

q= 1-0.35 = 0.65

P(X=7)= 0.35×0.65^7

0.35×0.0490

= 0.0172

4. P(x; μ) = (e^-μ) (μ^x) / x!

e= 2.7183

μ= 50/12 = 4.167

I. P(X=0) =

(2.7183^-4.167)(4.167^0)/0!

P(X=0) = 0.015498

~0.0155

ii. P(X=7) =

(2.7183^-4.167)(4.167^7)/7!

(0.015498×21815.4664)/5040

=0.06708

~0.067

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function meaning that the variable can only take specific values in a (potentially infinite) list Examples include number of mosquito bite on person,number of car accidents over a certain range of a road, number of computer crashes in a day etc.Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution.Uniform probability distribution are probability distribution with equally likely outcome. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

No chocolate

7 chocolates.

P(x=0) ×/0

=0.0067×1/1

=0.0067

P(x=7) ×/5040

= 0.0067/5040

=523.4/5040

=0.1038.

NAME: Edwin Chinedu Augustine

REG. NO: 2019/249508

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The formula is stated below:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: Otutu Chisom Judith

REG. NO: 2019/242963

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: Okeanyaego Victor Chidubem

REG. NO: 2019/244068

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: NSAN MANASSEH OSAMINEN

REG. NO: 2019/249517

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: Udeze Chibuike Kizito

REG. NO: 2019/249949

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME: Nebo Casmir Chukwuemeka

REG. NO: 2019/244263

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

ODUM PRECIOUS NAOMI

2019/241381

odumprecious001@gmail.com

1) a. Poisson distribution – This is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Therefore, Poisson distribution can be simply defined as a discrete distribution that measures the probability of a given number of events happening in a specified time or period.

b. Uniform probability distribution – This is a continuous probability distribution that is concerned with events that are equally likely to occur when working about problems that have a uniform distribution, we note if the data is inclusive or exclusive.

c. Exponential probability distribution – It is the probability distribution of the time between events in a Poisson point process. It has a memoryless property. It is used to model the time elapsed before a given event occurs.

2) 6 Doctors and 20 Nurses

7 names randomly selected without replacement

No of possible outcome = 26

P (4 doctors and 3 nurses) = 4/26 X 3/22 = 12/572 = 3/143

3) This implies failure would occur 7 times before she gets married (at the 8th trial)

Let X be the number of relationships before her marriage

P (X = X) = P (1 – P)^x or PQ^x

P (X = 7) = 0.35 x (0.65)^7

P (X = 7) = 0.0172

4) a. Lambda = 50, X = 0

P (X = x) = e^-lambda Lambda^x / x!

P (X = 0) = e^-50 . 50^0 / 0!

= e^-50 . 1 / 1 = 1.92875

= approximately 2

b. Lambda = 50, X = 7

P (X = 7) = e^-50 . 50^7 / 7!

= 1.9287498E-22 . 7.8125E11 / 5046

= 2.9861986

= approximately 3.

OKOH RACHEL IFUNANYA

2019/242735

okoh.rachel04@gmail.com

1. Write short notes on the following types of probability distributions:

a. Poisson probability distribution: Poisson probability distribution is a discreet probability distribution that gives the probability of an independent events occurring in a fixed time. Examples include:Number of car scratches on a car paint, number of accident that happened on a particular road, number of blue cars that passed a particular road etc.this is very useful to managers to know how customers feel about their product or services.

b. Uniform probability distribution: Uniform Probability distribution is a Continuous distribution in which all outcomes are equally likely.

c. Exponential probability distribution: This is also a continuous distribution that is usually present, when we are dealing with events that change rapidly. Example news, blogs.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

NAME: Ngana Thaddeus Ifeanyi

REG. NO: 2019/246750

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

1a. POISSON PROBABILITY DISTRIBUTION: The poisson distribution is used to check how many times an event is bound to occur.

The poisson distribution is also a discrete function meaning it can only take specific values in a potentially infinite list.

1a. UNIFORM PROBABILITY DISTRIBUTION: this is a type of probability that has two types : continuous and discrete, also in this probability all outcomes are equally likely to occur.

In continuous uniform probability outcomes are infinite while in discrete uniform probability all outcomes are equally likely and have finite values.

1c. EXPONENTIAL PROBABILITY DISTRIBUTION: of all the continuous distributions this is more popular and more widely used. It is the probability distribution of the time between events in a poisson point process.

2. rCx × (N-r)C(n-x)÷NCn

Where r=6

n=7

N=26

x=4

= 6C4 × (26-6)C(7-4)÷26C7

= 6C4 × 20C3÷26C7

=15×1140÷657800

=0.0260

3. pq^x

Where p=0.25

q=1-p =1-0.25 =0.65

x=7

= 0.25×0.65^7

= 0.25×0.0490

= 0.0172

4. P(X=x) = √^x × £^-2

Where √= lander=50/12 =4.17

£= exponatial

4a. Where x=0

P(X=x) =

4.17^0 × £^-4.17/0!

1 × 0.155/1

P(X=0) = 0.155

4b. Where x=7

P(X=7) =

4.17^7 × £^-4.17/7!

=21925.65 ×0.015/5040

= 338.80/5040

P(X=7)= 0.0672

NAME: EKWEKE DEBORAH ONYINYECHI

REG.NO: 2019/243791

Short Notes on:

Poisson Probability Distribution

This is a probability distribution that is used to show how many times an event occurred in a particular area of opportunity over a specified period of time. They involve independent events that occur at a constant rate within a given period of time. It is a discrete function (uses discrete numbers).

Uniform Probability Distribution

This refers to a type of probability distribution in which all outcomes are equally likely. For example, a deck of cards has within it uniform distributions because the likelihood of drawing a hat, a club, a diamond or a spade is equally likely. There are two types; Discrete Uniform Distribution Probability and Continuous Uniform Probability Distribution.

Exponential Probability Distribution:

This is the distribution of the time between events in a Poisson Point process, that is, a process in which events occur continuously and independently at a constant average rate. It answers such questions as:

How much long do we need to wait until a costumer enters our shop? How long will a piece of machinery work without breaking down?

2) 6 Doctors and 20 Nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 Doctors and 3 Nurses are picked?

FORMULAR:

/

N=26 r=6 n=7

The probability that x=4

/

6! / 4!(6-4)!

= 6*5*4! / 2! 4!

= 15

The probability that x=3

3) Using the Geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will actually get married in her 8th relationship.

P(X=x) =

P= 0.35 q=0.65 x=8

P(X=x) =

=

= 0.0172

4) To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same portion is used for all batches. If a cookie is chosen at random from a large batch, also using lambda (mean) to the nearest whole number. What is the probability that it contains:

a) no chocolate chips

/ x!

λ = 50/12= 4.2 x=0 e=2.7183

/ 0!

=0.015

b) 7 chocolate chips

/ 7!

=23053.9 * 0.015 / 5040

=0.0686

1a. POISSON PROBABILITY DISTRIBUTION: The poisson distribution is used to check how many times an event is bound to occur.

The poisson distribution is also a discrete function meaning it can only take specific values in a potentially infinite list.

1a. UNIFORM PROBABILITY DISTRIBUTION: this is a type of probability that has two types : continuous and discrete, also in this probability all outcomes are equally likely to occur.

In continuous uniform probability outcomes are infinite while in discrete uniform probability all outcomes are equally likely and have finite values.

1c. EXPONENTIAL PROBABILITY DISTRIBUTION: of all the continuous distributions this is more popular and more widely used. It is the probability distribution of the time between events in a poisson point process.

2. rCx × (N-r)C(n-x)÷NCn

Where r=6

n=7

N=26

x=4

= 6C4 × (26-6)C(7-4)/26C7

= 6C4 × 20C3/26C7

=15×1140/657800

=0.0260

3. pq^x

Where p=0.25

q=1-p =1-0.25 =0.65

x=7

= 0.25 × 0.65^7

= 0.25 × 0.0490

= 0.0172

4. P(X=x) = √^x × £^-2

Where √= lander=50÷12 =4.17

£= exponatial

4a. Where x=0

P(X=x) =

4.17^0 × £^-4.17÷0!

1 × 0.155÷1

P(X=0) = 0.155

4b. Where x=7

P(X=7) =

4.17^7 × £^-4.17÷7!

=21925.65 ×0.015/5040

= 338.80/5040

P(X=7)= 0.0672

1. Write short notes on the following types of probability distributions:

Poisson probability distribution: it is a discrete function used to show how many times an event is likely to occur over a specified period. Examples includes number of mosquito bite on person, number of motor accident in a day etc.

Uniform probability distribution: it is also known as Continuous uniform distribution . It is a family of symmetric probability distributions. This continuous probability distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a continuous probability distribution in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

4. To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) ×/0

=0.0067×1/1

=0.0067

P(x=7) ×/5040

= 0.0067/5040

=523.4/5040

=0.1038.

NAME:. Ezurueme Ogechi

REG NO: 2019/251620

DEPARTMENT: ECONOMICS

EMAIL ADDRESS: oezurueme@gmail.com

1A] POISSON PROBABILITY DISTRIBUTION

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. The French mathematician Siméon-Denis Poisson developed this function in 1830 to describe the number of times a gambler would win a rarely won game of chance in a large number of tries. It is a tool that helps to predict the probability of certain events happening when you know how often the event has occurred. It gives us the probability of a given number of events happening in a fixed interval of time.

B] UNIFORM PROBABILITY DISTRIBUTION

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A uniform distribution, also called a rectangular distribution, is a probability distribution that has constant probability.

This distribution is defined by two parameters, a and b:

a is the minimum.

b is the maximum.

C] EXPONENTIAL PROBABILITY DISTRIBUTION

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. In Probability theory and statistics, the exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens. It is a process in which events happen continuously and independently at a constant average rate. The exponential distribution has the key property of being memoryless. The exponential random variable can be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the supermarket follows an exponential distribution.

2. P (X = 4 doctors and 3 nurses)

N=26. n=7. r=6.

6C4 X 20C3 /26C7

= 15 X 1140

—————— =0.0259

657800

3. 0.35 X (0.65)⁷ = 0.017

4. Pr{X = x} = e−μ. μx/x!

a. μ = 12/50 = 0.24

x = 0

e– ·²⁴ (0.24)⁰ / 0! = 0.786

b. e– ·²⁴ (0.24)⁷ / 7! = 7.158

1a. POISSON PROBABILITY DISTRIBUTION: The poisson distribution is used to check how many times an event is bound to occur.

The poisson distribution is also a discrete function meaning it can only take specific values in a potentially infinite list.

1a. UNIFORM PROBABILITY DISTRIBUTION: this is a type of probability that has two types : continuous and discrete, also in this probability all outcomes are equally likely to occur.

In continuous uniform probability outcomes are infinite while in discrete uniform probability all outcomes are equally likely and have finite values.

1c. EXPONENTIAL PROBABILITY DISTRIBUTION: of all the continuous distributions this is more popular and more widely used. It is the probability distribution of the time between events in a poisson point process.

2. rCx × (N-r)C(n-x)÷NCn

Where r=6

n=7

N=26

x=4

= 6C4 × (26-6)C(7-4)÷26C7

= 6C4 × 20C3÷26C7

=15×1140÷657800

=0.0260

3. pq^x

Where p=0.25

q=1-p =1-0.25 =0.65

x=7

= 0.25 × 0.65^7

= 0.25 × 0.0490

= 0.0172

4. P(X=x) = √^x × £^-2

Where √= lander=50/12 =4.17

£= exponatial

4a. Where x=0

P(X=x) =

4.17^0 × £^-4.17÷0!

1 × 0.155÷1

P(X=0) = 0.155

4b. Where x=7

P(X=7) =

4.17^7 × £^-4.17÷7!

=21925.65 ×0.015÷5040

= 338.80÷5040

P(X=7)= 0.0672

NAME: OMEJE GOD’SFAVOUR CHIEMERIE

REG. NO: 2019/244972

Short Notes on the following:

Poisson Probability Distribution

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period of time. It can be used to estimate how likely it is that something will happen “X” number of times. It enables managers to introduce optimal scheduling systems that would not work with, say, a normal distribution.

Uniform Probability Distribution

Uniform distributions are probability distributions with equally likely outcomes. For example, a coin has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. They are of two types; Discrete and Continuous Uniform Distribution. In discrete uniform distribution, outcomes are discrete and have the same probability while in Continuous uniform distribution, outcomes are continuous and infinite.

Exponential Probability Distribution

This is a continuous probability distribution used to model the time elapsed before a given event occurs. Sometimes it is also called a negative exponential distribution. It answers questions like: How much time will elapse before an earthquake occurs in a given region? How long will it take before a call center receives the next phone call?

2) 6 Doctors and 20 Nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 Doctors and 3 Nurses are picked?

FORMULAR:

/

N=26 r=6 n=7

The probability that x=4

/

6! / 4!(6-4)!

= 6*5*4! / 2! 4!

= 15

The probability that x=3

/

3) Using the Geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will actually get married in her 8th relationship.

P(X=x) =

P= 0.35 q=0.65 x=8

P(X=x) =

=

= 0.0172

4) To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same portion is used for all batches. If a cookie is chosen at random from a large batch, also using lambda (mean) to the nearest whole number. What is the probability that it contains:

a) no chocolate chips

/ x!

λ = 50/12= 4.2 x=0 e=2.7183

/ 0!

=0.015

b) 7 chocolate chips

/ 7!

=23053.9 * 0.015 / 5040

=0.0686

NAME: ONU CHINECHEREM EXCELLENCE

REG NO: 2019/241446

DEPARTMENT: ECONOMICS

COURSE CODE: ECO 231

COURSE TITLE: INTERMEDIATE ECONOMIC STATISTICS I

ASSIGNMENT ON ECO 231

(1a) Poisson Probability distribution: This is used when an event occurs in a given area of opportunity; that is an interval of time or area where more than one event can occur.

(1b) Uniform Probability Distribution: In statistics, uniform probability distribution refers to a type of probability distribution in which all the outcomes of event are likely equal e.g. a coin.

(1c) exponential Probability Distribution: This is often concerned with the amount of time it takes for before a specific event occurs. i.e. the time between events and its occurrence continuously and independently constant rate.

(2) Using hypergeometric Distribution

P(x=x)

P(x = 4doctor and 3nurses)

Let N = 26

n = 7

r = 6

P(x = 7) =

=> = 6C4 x 20C3 = 15 x 1140

= 26C7 = 657800

= = 0.026

(3) Using Geometric Distribution

P(x = x) = P(1 – P)x

P(x = 7) = 0.35 (0.65)7

0.0172

(4) Using Poisson Distribution

P(x = x) =

e = 2.71828

= 50/12 = 4.17 4

0.01832

e = 2.71828

= 50/12 = 4.17 4

0.0595

ASSIGNMENT ON ECO 231 Date: 23/11/2021The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

Answers to question 1: (A) Poisson Probability Distribution?

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

USES OF POISSON PROBABILITY DISTRIBUTION

1)Poisson distributions are used by businessmen to make forecasts about the number of customers or sales on certain days or seasons of the year.

2)With the Poisson distribution, companies can adjust supply to demand in order to keep their business earning good profit. In addition, waste of resources is prevented.

3)By using this tool, businessmen are able to estimate the time when demand is unusually higher, so they can purchase more stock.

(B) The Uniform Probability Distribution

Uniform distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. The probability is constant since each variable has equal chances of being the outcome.

Example of Uniform Distribution

If you stand on a street corner and start to randomly hand a $100 bill to any lucky person who walked by, then every passerby would have an equal chance of being handed the money. The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. It would not be described as uniform probability.

Types of Uniform Distribution

1) Discrete uniform distribution

In statistics and probability theory, a discrete uniform distribution is a statistical distribution where the probability of outcomes is equally likely and with finite values.

2)Continuous uniform distribution

Not all uniform distributions are discrete; some are continuous. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. Unlike discrete random variables, a continuous random variable can take any real value within a specified range.A continuous uniform distribution usually comes in a rectangular shape. A good example of a continuous uniform distribution is an idealized random number generator. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. However, there is an infinite number of points that can exist.

Mean (A + B)/2

Median (A + B)/2

Range B – A

Standard Deviation (B−A)212−−−−−√

Coefficient of Variation (B−A)3√(B+A)

Skewness 0

Kurtosis 9/5

(C) Exponential Probability distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.The exponential distribution refers to the continuous and constant probability distribution which is actually used to model the time period that a person needs to wait before the given event happens and this distribution is a continuous counterpart of a geometric distribution that is instead distinct.

A continuous random variable X is said to have an exponential distribution with parameter λ>0, shown as X∼Exponential(λ), if its PDF is given by

fX(x)={λe−λx0x>0otherwise.

Answer to question 2:

N=26,n=7,x=(4 doctors and 3 nurses),r=6

P(X=x)= (6C4)(20C3)/(26C7)=0.0259

Answer to question 3:

P(X=x)= p(1-p)*

0.35(0.65)^7=0.01716

Answer to question 4a: Using poisson formula, we get 0.00000000000000000000019288

4b) we get 0.00000000000002989

REG No: 2017/250122

Name: Omeje Jacinta ukamaka

Economics major

1a. Poisson probability distribution: this is a statistical instrument that’s helps to tell the probability of some events occurring when we know how often the event has occurred, that’s to say, poisson distribution gives the probability of a given number of events occurring at a fixed interval of time.

The poisson probability distribution is a discrete function and this means that it’s variables can take finite values such as values like 0, 1, 2, 3, 4 etc, without decimals or fractions.

The formula for the Poisson probability distribution is given as: P(X=x)=(λ^x × e^-λ)/x!. Where;

e is the Euler’s number

x is the number of occurrence

λ is equal to its mean and variance

x! Is the factorial of x

The Poisson distribution can also be useful in biology, finance and any other situation in which events are time-independent.

1b. Uniform probability distribution: in statistics, this is a type of continuous distribution that describes the probability of events occurring equally. It’s also called rectangular distribution. It has a constant probability and it’s defined by two parameters, a and b. Where;

a is the minimum and

b is the maximum

There are two types of uniform probability distribution namely: the discrete uniform probability distribution and the continuous uniform probability distribution.

In a discrete uniform probability distribution the outcomes of events finite and have the same probability for instance, the probability of tossing a coin will result in either a head or tail and its probability is the same. While in a continuous uniform probability distribution the outcomes of events are infinite and it can range between 0.0 and 0.1 with an equal chance of occurring.

The formula for the uniform probability distribution is given as: P(X=x)=1/b-a

1c. Exponential probability distribution: this is a continuous distribution that deals with the amount of time before an event occurs that is, it’s used to measure the expected time for an event to occur. The formula is given as: P(X=x)= λe^-λx.

2. P(X=x)= [(rCx)(N-rCn-x)]/NCn

Where r= 6, N= 26, n= 7 and x= 4 doctors and 3 nurses.

P(X=4 doctors and 3 nurses)= (6C4 × 20C3)/26C7

P(X=4 doctors and 3 nurses)= 0.0259

3. P(X=x)= P(1-p)^x

Where p= 0.35, 1-p= 0.65, x= 7

P(X=x)= 0.35(0.65)^7

P(X=x)= 0.017157

4a. P(X=x)= (λ^x × e^-λ)/x!

Where λ= 50/12 = 4.12 ≈ 4

e= 2.7183

x= 0

P(X=0)= (4^0 × 2.7183^-4)/0!

P(X=0)= (1×0.00183)/1

P(X=0)= 0.00183

4b. P(X=x)= (λ^x × e^-λ)/x!

Where λ= 4, e= 2.7183, x=7

P(X=7)= (4^7×2.7183^-4)/7!

P(X=7)= (16384×0.00183)/5040

P(X=7)= 0.00595

1a.) A Poission Probability Distribution; These was conceptualized by french mathematician simeon denis poisson, it is statistic theorem that explain how many times an occasion is likely to occur over a specified period.

1b.) Uniform Probability Distribution; It is most common and understandable distribution. It is type distribution that show all possible outcome has an equal probability of happening or occurring.

1c.) Exponential Probability Distribtion; In statistic, it is type of distribution is defined as the probability distribution of time between events in the poisson point process.

2.)

r =6 N=26

N-r = 20

n =7, x=4, n-x=3

3.) Using formula for geometric distribution

Where x=n-1 and q= 1-p

Therefore

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

4.)

When x=0

When x=7

POISSON DISTRIBUTION

Poisson Probability Distribution is another Discrete Random Variable. The Poisson Distribution is used to measure Time,Frequency,and Rate. In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

The Greek Letter Lamba(λ) is called the parameter of distribution which represents average number of times the Random Variable occurs in a specified interval or Space.

A discrete random variable is said to have a Poisson distribution, with parameter λ>0. It has a probability mass function given by:

P(X=x)=(λ^x × e^-λ)/x!

where:

x is the number of occurrences

e is Eulers number (e=2.7183)

! is the factorial function.

UNIFORM DISTRIBUTION

In probability theory and statistics, the uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)). Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution.The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution’s support.

The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself (but it is dependent on the interval size), so long as the interval is contained in the distribution’s support.

It has a probability mass function given by:

P(X=x)=1/b-a

EXPONENTIAL PROBABILITY DISTRIBUTIONIn probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

The exponential distribution is sometimes parametrized in terms of the scale parameter β = 1/λ,which is also the mean.The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by. E(X)=1/λ,.

The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant failure rate.

P(X=x)= λe^-λx

2.

Six(6) doctors and 20 nurses attended a small conference,26 names are put In a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P (4 docs and 3 nurses): P(X=4D) × P(X=4N)

For P(X=4D)

r=6

n=7

N=26

x= 4 doctors

C = combination

D = Doctors

P(X=x)= [rCx] . [N-rCn-x]/NCn

= (6C4)(20C3)/26C7

=(15 × 1140)/657800

=17100/657800

=0.026.

For P(X=3N)

r=20; n=7; N=26 ; x= 3 Nurses; C = combination; N = Nurse

P(X=x)= [rCx] . [N-rCn-x]/NCn

= (20C3)(6C4)/26C7

=(15 × 1140)/657800

=17100/657800

=0.026.

Therefore, P (4 docs and 3 nurses) = 0.026 × 0.026

= 0.000676

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8 relationship.

Formula for geometric distribution; pq^x

Given

P=0.35, x= 8-1 = 7, q = 1- 0.35 = 0.65

P(x=8) = (0.35) (0.65)^7

=(0.35)(0.049)

P(x=8) = 0.0176.

4.

To make a dozen chocolate chip cookies, 50 chocolate chips are mixed into the dough, the same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lambda to the nearest whole number, what is the probability that it contains?

No chocolate

7 chocolate

Formula for Poisson probability distribution

P(X=x)=(λ^x × e^-λ)/x!

Given

λ = 50/12 = 4.16

λ = 4( to the nearest whole number)

e = 2.718

(a). P(x=0) ={2.718^⁻⁴ × 4^0}/0!

={0.0183 × 1}/1

=0.0183

b. P(x=7) = 2.718^⁻⁴ × 4^⁷ /7!

={0.0183 × 16384}/5040

=299.8/5040

=0.0594

1a) Poison probability distribution is a probability of number of times an event occurs in a given area of opportunity. There can be unlimited number of possible outcomes unlike the binomial probability distribution of only two possible outcomes (success or failure). Examples : the number of injuries in a working place, the number of things sold at a time in a shop.

1b) Uniform probability distribution is a distribution that has equal likely possible outcomes. There are two types of uniform probability distribution ; discrete and continuous. Examples are deck of cards and coins.

1c) Exponential probability distribution is a distribution in which events occur continuously and independently at a constant average rate. It is used for poison point processes.

2) using hypergeometric p(X=x) =( rCx) (N-rCn-x) ÷(NCn)

N = 6+20=26

n = 7

r = 6

P(X= 4 doctors and 3 nurse) =( 6C4) (26-6C7-4)÷ 26C7

= (6C4) (20C3) ÷ 26C7; (15)(1140) ÷ (657800)

= 0.026 answer

3) using geometric distribution p(X=x) = p(1-p)^x or pq^x

P = 0.35, q= 1- 0.35=0.65, x= 7

P(X=3) = 0.35 × (0.65) ^7

=0.0172 answer

4) using poison distribution p(X=x)= e^-h×h^x ÷ x!

e = 2.7183 ( constant), h= mean 1 ( to the nearest whole number), x=0

4a) P(X=0)= 2.7183^-1×1^0÷0!

= 0.3679 answer

4b) p(X=7)= 2.7183^-1×1^7÷7!

= 0.00007299 answer

EDWIN-UGODU STEPHEN CHIDI

2019/251264 (ECONOMICS)

INTERMEDIATE ECONOMIC STATISTICS (ECO 231)

ASSIGNMENT: 1. Write short note on the following types of probability distribution:

(a) Poisson probability distribution: this type of probability distribution is best when the number of times an event occurs in a given area of opportunity is of interest.

(b) Uniform probability distribution: sometimes called the bell curve, is a distribution that occurs naturally in many situation, and it’s symmetrical around its mean.

(c) Exponential probability distribution: treats cases of frequently or rapidly changing events like trends or news blog etc.

2. Six(6) doctors and twenty(20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Soln

Hypergeometric probability distribution function:

P(X=x) =(xCr)×(N-xCn-r)÷(NCn) ~combination~

Where: N=6+20=26, n=7, x=6 and r= 4

P(X=4) = (6C4)×(20C3)÷(26C7)

P(X=4) =0.026 approx.

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

Soln

Geometric probability distribution function: P(X=x) = p.q^x

where: x= n -1, P=0.35 and q=0.65

P(X=7) = (0.35)(0.65)^7 = 0.01716 approx.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using Lambda (the mean) to the nearest whole number. What is the probability that it contains:

a] No chocolate chips? Soln

Poisson probability function: P(X=x)= e^-Y.Y^x/x!

where: e=2.718, Y(lambda)=50/12=4.1667 and x=0

P(X=0)= (e^-4.1667×4.1667^0)/0! = e^-4.1667 = 0.0155 approx.

(b) 7 chocolate chip? Soln

P(X=7)= (e^-4.1667×4.1667^7)/7!= 338.042/5040= 0.067 approx.

NAME: OMEBE SAMUEL OFORBUIKE

REG NO: 2019/246454

Write short notes on:

Poisson Distribution

The poison distribution is a theoretical discrete probability distribution that is very useful in situations where the discrete events occur in a continuous manner. This has a huge application in many practical scenario like determining the number of calls received per minute at a call centre or the number of unbaked cookies in a batch at a bakery and much more.

Uniform Distribution

A discrete uniform probability distributions one in which all elementary events in the sample space have an equal opportunity of occurring. As a result, for a finite sample space of size n, the probability of an elementary event occurring is 1/n.

Exponential Distribution

The exponential probability (also called the negative exponential probability) is a probability distribution that describes time between events in a poison process. There is a strong relationship between the poison distribution and the exponential distribution. For example let’s say a poison distribution models the number of birth in a given period.

Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

NAME: Ucheama Calista Ngozi

Reg No; 2019/243039

1. Write short notes on:

*Poisson Probability Distribution*

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading.

*Uniform Probability Distribution*

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints.

*Exponential Probability Distribution*

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Solution3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

1a) Poison probability distribution is a probability of number of times an event occurs in a given area of opportunity. There can be unlimited number of possible outcomes unlike the binomial probability distribution of only two possible outcomes (success or failure). Examples : the number of injuries in a working place, the number of things sold at a time in a shop.

1b) Uniform probability distribution is a distribution that has can have equal possible outcomes. There are two types of uniform probability distribution ; discrete and continuous. Examples are deck of cards and coins.

1c) Exponential probability distribution is a distribution in which events occur in a continuously and independent manner at a constant average rate. It is used for poison point processes.

2) using hypergeometric p(X=x) =( rCx) (N-rCn-x) ÷(NCn)

N = 6+20=26

n = 7

r = 6

P(X= 4 doctors and 3 nurse) =( 6C4) (26-6C7-4)÷ 26C7

= (6C4) (20C3) ÷ 26C7; (15)(1140) ÷ (657800)

= 0.026 answer

3) using geometric distribution p(X=x) = p(1-p)^x or pq^x

P = 0.35, q= 1- 0.35=0.65, x= 7

P(X=3) = 0.35 × (0.65) ^7

=0.0172 answer

4) using poison distribution p(X=x)= e^-h×h^x ÷ x!

e = 2.7183 ( constant), h= mean 1 ( to the nearest whole number), x=0

4a) P(X=0)= 2.7183^-1×1^0÷0!

= 0.3679 answer

4b) p(X=7)= 2.7183^-1×1^7÷7!

= 0.00007299 answer

1a) Poison probability distribution is a probability of number of times an event occurs in an area of concentration. There can be unlimited number of possible outcomes unlike the binomial probability distribution of only two possible outcomes (success or failure). Examples : the number of injuries in a working environment, the number of things sold at a time in a shop.

1b) Uniform probability distribution is a distribution that has can have equal possible outcomes. There are two types of uniform probability distribution ; discrete and continuous. Examples are deck of cards and coins.

1c) Exponential probability distribution is a distribution in which events occur in a continuously and independent manner at a constant average rate. It is used for poison point processes.

2) using hypergeometric p(X=x) =( rCx) (N-rCn-x) ÷(NCn)

N = 6+20=26

n = 7

r = 6

P(X= 4 doctors and 3 nurse) =( 6C4) (26-6C7-4)÷ 26C7

= (6C4) (20C3) ÷ 26C7; (15)(1140) ÷ (657800)

= 0.026 answer

3) using geometric distribution p(X=x) = p(1-p)^x or pq^x

P = 0.35, q= 1- 0.35=0.65, x= 7

P(X=3) = 0.35 × (0.65) ^7

=0.0172 answer

4) using poison distribution p(X=x)= e^-h×h^x ÷ x!

e = 2.7183 ( constant), h= mean 1 ( to the nearest whole number), x=0

4a) P(X=0)= 2.7183^-1×1^0÷0!

= 0.3679 answer

4b) p(X=7)= 2.7183^-1×1^7÷7!

= 0.00007299 answer

1a) Poison probability distribution is a probability distribution of how an event occurs in number of times . It has no limited possible outcomes unlike the binomial probability distribution Example : the number of things sold at a time in a shop.

1b) Uniform probability distribution is a distribution that the possible outcomes can have equal occurrence and there are of two types ; discrete and continuous. Exampleare coins tossed.

1c) Exponential probability distribution is a distribution in which events occur in a constant and free manner at a constant average rate.

2) using hypergeometric p(X=x) =( rCx) (N-rCn-x) ÷(NCn)

N = 6+20=26

n = 7

r = 6

P(X= 4 doctors and 3 nurse) =( 6C4) (26-6C7-4)÷ 26C7

= (6C4) (20C3) ÷ 26C7; (15)(1140) ÷ (657800)

= 0.026 answer

3) using geometric distribution p(X=x) = p(1-p)^x or pq^x

P = 0.35, q= 1- 0.35=0.65, x= 7

P(X=3) = 0.35 × (0.65) ^7

=0.0172 answer

4) using poison distribution p(X=x)= e^-h×h^x ÷ x!

e = 2.7183 ( constant), h= mean 1 ( to the nearest whole number), x=0

4a) P(X=0)= 2.7183^-1×1^0÷0!

= 0.3679 answer

4b) p(X=7)= 2.7183^-1×1^7÷7!

= 0.00007299 answer

NAME: Monroe Favour Chibuzor

REG. NO: 2019/249605

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time from now until an earthquake occurs has an exponential distribution. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The formula is stated below:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqxBefore she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

1a) Poisson probability distribution is a probability distribution of how an event occurs in number of times . It has no limited possible outcomes unlike the binomial probability distribution Example : the number of things sold at a time in a shop.

1b) Uniform probability distribution is a distribution that the possible outcomes can have equal occurrence and there are of two types ; discrete and continuous. Exampleare coins tossed.

1c) Exponential probability distribution is a distribution in which events occur in a constant and free manner at a constant average rate.

2) using hypergeometric p(X=x) =( rCx) (N-rCn-x) ÷(NCn)

N = 6+20=26

n = 7

r = 6

P(X= 4 doctors and 3 nurse) =( 6C4) (26-6C7-4)÷ 26C7

= (6C4) (20C3) ÷ 26C7; (15)(1140) ÷ (657800)

= 0.026 answer

3) using geometric distribution p(X=x) = p(1-p)^x or pq^x

P = 0.35, q= 1- 0.35=0.65, x= 7

P(X=3) = 0.35 × (0.65) ^7

=0.0172 answer

4) using poison distribution p(X=x)= e^-h×h^x ÷ x!

e = 2.7183 ( constant), h= mean 1 ( to the nearest whole number), x=0

4a) P(X=0)= 2.7183^-1×1^0÷0!

= 0.3679 answer

4b) p(X=7)= 2.7183^-1×1^7÷7!

= 0.00007299 answer

1a.Poisson probability distribution:it is used to show how many times and event is likely to occur over a specified period.They are often used by to understand independent events that occurs at a constant rate within a given interval of time.it is a discrete function, ie the variable can only take specific values in a list.

b. Uniform probability distribution: are probability distribution with equal likely outcome.It is of two types namely discrete and continuous whose outcomes are the same and infinite.Respectively.

c. Exponential probability distribution: is often concerned with the amount of time until some specific event occurs.

2. The probability = 0.2206

3. The probability = 0.01712

4. a.P(X=0)= 0.0183

b.P(X=7)= 0.0595

EDWIN-UGODU STEPHEN CHIDI

2019/251264 (ECONOMICS)

INTERMEDIATE ECONOMIC STATISTICS (ECO 231)

ASSIGNMENT: 1. Write short note on the following types of probability distribution:

(a) Poisson probability distribution: this type of probability distribution is best when the number of times an event occurs in a given area of opportunity is of interest.

(b) Uniform probability distribution: sometimes called the bell curve, is a distribution that occurs naturally in many situation, and it’s symmetrical around its mean.

(c) Exponential probability distribution: treats cases of frequently or rapidly changing events like trends or news blog etc.

2. Soln

Hypergeometric probability distribution function:

P(X=x) =(xCr)×(N-xCn-r)÷(NCn) ~combination~

Where: N=6+20=26, n=7, x=6 and r= 4

P(X=4) = (6C4)×(20C3)÷(26C7)

P(X=4) =0.026 approx.

3. Soln

Geometric probability distribution function: P(X=x) = p.q^x

where: x= n -1, P=0.35 and q=0.65

P(X=7) = (0.35)(0.65)^7 = 0.01716 approx.

4. Soln

(a) no chocolate chips? Soln

Poisson probability function: P(X=x)= e^-Y.Y^x/x!

where: e=2.718, Y(lambda)=50/12=4.1667 and x=0

P(X=0)= (e^-4.1667×4.1667^0)/0! = e^-4.1667 = 0.0155 approx.

(b) 7 chocolate chip? Soln

P(X=7)= (e^-4.1667×4.1667^7)/7!= 338.042/5040= 0.067 approx.

NAME: UGAH CHIKAODILI UDODILI.

DEPARTMENT: ECONOMICS.

REGISTRATION NUMBER: 2019/243002.

1a.) Poisson Probability Distribution: These was conceptualized by French mathematician, Simeon Denis Poisson, it is a statistic theorem that explain how many times an occasion is likely to occur over a specified period.

1b.) Uniform Probability Distribution: It is the most common and understandable distribution. It is a type of distribution that shows all possible outcomes and has an equal probability of happening or occurring.

1c.) Exponential Probability Distribution: In statistics, it is a type of distribution that is defined as the probability distribution of time between events in the Poisson point process.

2.)

r =6 N=26

N-r = 20

n =7, x=4, n-x=3

3.) Using formula for geometric distribution

Where x=n-1 and q= 1-p

Therefore

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

4.)

When x=0

When x=7

ELIGWEDIRE VICTOR OZIOMA

2019/249216 (ECONOMICS)

INTERMEDIATE ECONOMIC STATISTICS (ECO 231)

1.(a) Poisson probability distribution: this type of probability distribution is best when the number of times an event occurs in a given area of opportunity is of interest.

(b) Uniform probability distribution: sometimes called the bell curve, is a distribution that occurs naturally in many situation, and it’s symmetrical around its mean.

(c) Exponential probability distribution: treats cases of frequently or rapidly changing events like trends or news blog etc.

2. Hypergeometric: P(X=x) =(xCr)×(N-xCn-r)÷(NCn)

Where: N=6+20=26, n=7, x=6 and r= 4

P(X=4) = (6C4)×(20C3)÷(26C7)

P(X=4) =0.026 approx.

3. Geometric probability: P(X=x) = p.q^x

where: x= n -1, P=0.35 and q=0.65

P(X=7) = (0.35)(0.65)^7 = 0.01716 approx.

4. a] No chocolate chips?Poisson : P(X=x)= e^-Y.Y^x/x!

where: e=2.718, Y(lambda)=50/12=4.1667 and x=0

P(X=0)= (e^-4.1667×4.1667^0)/0! = e^-4.1667 = 0.0155 approx.

(b) 7 chocolate chip? SolnP(X=7)= (e^-4.1667×4.1667^7)/7!= 338.042/5040= 0.067 approx.

1a

Poisson probability distribution:It is a discrete probability distribution of a given number of events occurring in a fixed interval of time or space.

1b

Uniform probability distribution: The distribution describes an experiment where there is an arbitrary outcome that lies between bounds. The bounds are defined by parameters.

1c

Exponential probability distribution:The probability distribution of the time between events.

2

The probability of picking 4 doctors without replacement =

4÷26 + 4÷25 + 4÷24 + 4÷23= 0.15+0.16+0.16+0.17=0.64

The probability of picking 3 nurses without replacement =

3÷26 + 3÷25 + 3÷24=0.12+0.12+0.13=0.37

3

P=0.35, q=1-0.35= 0.65, x=1-8= 7

pq^x

0.35×0.65^7

=0.0172

4a

P(x=x) =/\^x.e^-/\÷x!

=4.17^0×e^-4.17÷0

=0.0155

4b

P(x=x) =/\^x.e^-/\÷x!

=4.17^7×e4.17^7÷7!

=0.65

Name: Chidubem Joshua

Reg no:2019/244235

Email: joshuadubem71@gmail.com

UCHEOMA DANIELLA CHIMDINDU

(2019/241763)

Danympompo123@gmail.com

ECO 231 ECONOMICS STATISTICS

1. (a) POISSON PROBABILITY DISTRIBUTION

This is a probability distribution of a discrete random variable that stands for the number (count) of statistically independent events occurring within a unit of time or space. The probability function (A mathematical function giving the probability occurrence to different outcomes of an experiment) is defined as;

f(x) =P(X=x)=(e^(-λ).λ^x)/x!, f(x) = ∑_(k=0)^(k=x)▒〖f(k)〗, x = 0,I,2…

Where;

x = Number of events in an area of opportunity

𝝀 = Expected number of events

e = Base of the natural logarithm system (2.71828…)

This Distribution was introduced by Simeon Denis Poisson when he discussed the wrongful conviction of prisoners in a given country by focusing on certain random variables

that count the number of discrete occurrences (events/arrivals) that take place during a time interval of a given length. It has a strong theoretical background and very wide spectrum of practical applications.

The Poisson Probability Distribution is a special case of Binominal Distribution. In some situations, the former one can be used to approximate the latter one

(b). UNIFORM PROBABILITY DISTRIBUTION

The uniform distribution (continuous) is one of the simplest probability distributions in statistics. It is a continuous distribution, this means that it takes values within a specified range, e.g. between 0 and 1. A probability experiment with a uniform probability distribution has a sample space (set of all possible outcomes) for which each value is equally likely. That is, if there are 𝑛 possible values then 𝑃(𝑋 = 𝑥) = 1/n for every 𝑥. It’s important to understand that this doesn’t mean that each outcome is equally likely, but rather that each value is equally likely.

The probability function is;

F(x)= {█(1/(b-a) @0)┤

For a ≤ x ≤ b

For x b

(c). EXPONENTIAL PROBABILITY DISTRIBUTION

The exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. An interesting feature of the Poisson and Exponential distribution is that if Poisson provides an appropriate description of the number of occurrence per interval of time, then the Exponential distribution will provide a description of the length of tome between the occurrence. The Exponential distribution is usually present when we are dealing with events that are rapidly changing early; for example, blogs

3. This implies that failure occur 7 times before she got married in her 8th relationship.

Therefore, let X be the number of her relationships before her marriage,

The probability function is P(X = x) = p(1-p)x or pqx

P = 0.35 and q = 1- 0.35 = 0.65

P(X = 7) = 0.35*(0.65)7 = 0.01716 (5d.p)

4 P(X = x) = (e^(-λ).λ^x)/x!

X is the number of chocolate chips

To get 𝝀, 50/12 = 4 (to the nearest whole number)

e = 2.71828

(a) No chocolate chips

x = 0

P(X = 0) = (e-4.4o)/0! = (0.01832 *1)/1 = 0.01832

(b) 7 chocolate chips

x = 7

P(X = 7) = (e-4.47)/7! = (0.01832 * 16384)/5040 = 0.05955

NAME: UGWU SILAS CHINAZAEKPERE

DEPARTMENT: ECONOMICS

REGISTRATION NUMBER: 2019/244182

1a. Poisson Probability Distribution: This was conceptualized by French mathematician, by name Simeon Denis Poisson, it is a statistic theorem that explain how many times an occasion is likely to occur over a specified period.

1b. Uniform Probability Distribution: This is the most common and understandable distribution. It is a type of distribution that shows all possible outcomes and has an equal probability of happening or occurring.

1c. Exponential Probability Distribution: Exponential probability distribution in statistics, is a type of distribution that is defined as the probability distribution of time between events in the Poisson point process.

2. r =6 N=26 N-r = 20 n =7, x=4, n-x=3

3. Using formula for geometric distribution where x=n-1 and q= 1-p Therefore P = 0.35n =8 Therefore x= 8-1 =7 q=1-p=1-0.35=0.654.) When x=0 When x=7

NAME: NWOKAFOR CHIDERA CLARE

REG NO: 2019/249161

E-MAIL: nwokaforchidera49@gmail.com

ANSWERS

1a). POISSON PROBABILITY DISTRIBUTION is used to show how many times an event is likely to occur over a specified period. This means it is a count distribution. It is often used to understand independent events that occur at a constant rate within a given interval of time. It is a discrete function, meaning that the variable can only take specific values in a list. The formula is P(X=x)=(λ^x × e^-λ)/x!

1b). UNIFORM PROBABILITY DISTRIBUTION refers to a type of probability distribution in which all outcomes are equally likely. A coin has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. There are two types of uniform probability distribution; discrete and continuous uniform distribution. The formula is P(X=x)=1/b-a

1c). EXPONENTIAL PROBABILITY DISTRIBUTION is the probability distribution of time between events in the poisson point process. It is the continuous analogue of the geometric distribution. It concerns the amount of time until some specific events happens. The formula is P(X=x)= λe^-λx

2).

The probability of picking 4 doctors:

N= 26

n= 7

r= 6

x= 4

P(X=4) = 6C4x20C3/26C7

= 17100/657800

P(X=4) = 0.0260

The probability of picking 3 nurses:

x= 7-3=4

Since P(X=4) = 0.0260, we multiply both of them i.e: 0.0260×0.0260 = 0.000676

This is because we are asked to find probability of picking 4 doctors “and” 3 nurses.

3). P(X=x) = P(1-P)^x

P= 0.35

q= 1-P = 1-0.35 = 0.65

x= 8 = 8-1 = 7

P(X=7)= 0.35(0.65)^7

= 0.35×0.049

= 0.01715

4). P(X=x)=(λ^x × e^-λ)/x!

I). No chocolate chips:

λ= 50/12= 4.17= 4 to the nearest whole number

x= 0

P(X=x) = 2.718^-4(4)^0/0!

= 0.0183×1/1

= 0.0183

ii). 7 chocolate chips:

λ= 50/12= 4.17= 4 to the nearest whole number

x= 7

P(X=x) = 2.718^-4(4)^7/7!

= 0.0183×16384/5040

= 1332.0192/5040

= 0.26428

Nwabuebo Success Ekene

2019/248711

1a

Poisson probability distribution:It is a discrete probability distribution of a given number of events occurring in a fixed interval of time or space.

1b

Uniform probability distribution: The distribution describes an experiment where there is an arbitrary outcome that lies between bounds. The bounds are defined by parameters.

1c

Exponential probability distribution:The probability distribution of the time between events.

2

The probability of picking 4 doctors without replacement =

4÷26 + 4÷25 + 4÷24 + 4÷23= 0.15+0.16+0.16+0.17=0.64

The probability of picking 3 nurses without replacement =

3÷26 + 3÷25 + 3÷24=0.12+0.12+0.13=0.37

3

P=0.35, q=1-0.35= 0.65, x=1-8= 7

pq^x

0.35×0.65^7

=0.0172

4a

P(x=x) =/\^x.e^-/\÷x!

=4.17^0×e^-4.17÷0

=0.0155

4b

P(x=x) =/\^x.e^-/\÷x!

=4.17^7×e4.17^7÷7!

=0.65

Poisson probability distribution, In statistics a poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distribution are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Simèon Denis Poisson.

Uniform probability distribution, in statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distribution because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin is the same.

Exponential probability distribution, in a probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a poisson point process i.e a process in which event occur continuously and independently at a constant average rate it is a particular case of the gamma distribution, it is the continuous analogue of the geometry distribution, and it has the key property of being memory less in addition to being used for the analysis of Poisson point process it is found in various other context.

Where r=6

n=7

N=26

x=4

= 6C4 × (26-6)C(7-4)÷26C7

= 6C4 × 20C3÷26C7

=15×1140÷657800

=0.0260

3. pq^x

Where p=0.25

q=1-p =1-0.25 =0.65

x=7

= 0.25×0.65^7

= 0.25×0.0490

= 0.0172

4. P(X=x) = √^x × £^-2

Where √= lander=50/12 =4.17

£= exponential

4a. Where x=0

P(X=x) =

4.17^0 × £^-4.17/0!

1 × 0.155/1

P(X=0) = 0.155

4b. Where x=7

P(X=7) =

4.17^7 × £^-4.17/7!

=21925.65 ×0.015/5040

= 338.80/5040

P(X=7)= 0.0672

Dike Nwachukwu Onyedikachukwu 2019/241349

1a. Poisson probability distribution is used when you want to find out the number of times and event occurs in a given area of opportunity. It is used in testing how unusual an event frequency is for a given interval.

B. Uniform probability distribution is a type of distribution where all the outcomes of an event are equally likely to occur.

C. Exponential probability distribution is usually used when dealing with events that are rapidly changing early.

2. P(x=4 and 3)

N=26 n=7 r=6

P(x=4 and 3)= (6C4 × 20C3) ÷ (26C7)

= (15 × 1140) ÷ 657800

=17100 ÷ 657800

= 0•026

3. P=0•35 q=0•65 8-1 =7

=0•35 × (0•65^7)

=0•35 × 0•049

=0•01715

4a. Lambda = 50÷12

= 4 (approximately)

P(x=0)= (2•7183^-4 × 4^0) ÷ 0!

= (0.0183 × 1) ÷ 1

=0•0183

b. P(x=7) = (2•7183^-4 × 4^7) ÷ 7!

= (0•0183 × 16384) ÷ 5040

=0•0595

POISSON PROBABILITY DISTRIBUTION

In probability theory and statistics, the Poisson distribution named after French mathematician Siméon Denis Poisson, is a discrete probability distSoribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

UNIFORM PROBABILITY DISTRIBUTION

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)).[2] Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable.

EXPONENTIAL PROBABILITY DISTRIBUTIONIn probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65⁷

= 0.0176.

No chocolate

7 chocolates.

P(x=0) 2.7183–⁵×5⁰/0

=0.0067×1/1

=0.0067

P(x=7) 2.7183-⁷×5⁷/5040

= 0.0067×78125/5040

=523.4/5040

=0.1038

POISSON PROBABILITY DISTRIBUTION

In probability theory and statistics, the Poisson distribution named after French mathematician Siméon Denis Poisson, is a discrete probability distSoribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

UNIFORM PROBABILITY DISTRIBUTION

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)).[2] Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable.

EXPONENTIAL PROBABILITY DISTRIBUTIONIn probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65⁷

= 0.0176.

No chocolate

7 chocolates.

P(x=0) 2.7183–⁵×5⁰/0

=0.0067×1/1

=0.0067

P(x=7) 2.7183-⁷×5⁷/5040

= 0.0067×78125/5040

=523.4/5040

=0.1038

A-The Poisson distribution is a discrete probability function that means the variable can only take specific values in a given list of numbers, probably infinite. A Poisson distribution measures how many times an event is likely to occur within “x” period of time. In other words, we can define it as the probability distribution that results from the Poisson experiment. A Poisson experiment is a statistical experiment that classifies the experiment into two categories, such as success or failure. Poisson distribution is a limiting process of the binomial distribution.

B-Uniform distributions are probability distributions with equally likely outcomes. In a discrete uniform distribution, outcomes are discrete and have the same probability. … The frequency of occurrence decreases the farther you are from the mean in a normal distribution.The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. The probability density function is f(x)=1b−a f ( x ) = 1 b − a for a ≤ x ≤ b.

C-the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts.

2-Six [6] doctors and twenty [20] nurses attended a small conference all 26 names are put in a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3-Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

4-To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

NAME: UGWOKE MICHAEL-MARY IKECHUKWU

DEPARTMENT: ECONOMICS

REGISTRATION NUMBER: 2019/248716

1a.) Poisson Probability Distribution: Poisson Probability distribution shows how many times an event is likely to occur over a specific period of time. It helps to predict the probability of an event happening when you know how often the e vent has occur.

These was conceptualized by French mathematician, Simeon Denis Poisson

1b.) Uniform Probability Distribution: Uniform distribution is type of probability distribution in which all the outcomes are equally likely.

For instance, when a coin is tossed, the chance of getting a tail or head is the same

1c.) Exponential Probability Distribution: This is a continuous distribution that is commonly used to measure the expected time for a event to occur. It is often used to model the time elapsed between events

2.)

r =6 N=26

N-r = 20

n =7, x=4, n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.) Using formula for geometric distributionP(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261≈0.05954

NAME: ANYANWU PASCHAL OSITADINMA DEPARTMENT: ECONOMICS REGISTRATION NUMBER: 2019/244008 (1)A. POISSON PROBABILITY DISTRIBUTION: Poisson Probability distribution displays how many times an event tends to occur over a specific period of time. It presumes the probability of an event happening when knowledge of the event’s occurance has been made known. This was conceptualized by the French mathematician, Simeon Denis Poisson.B. UNIFORM PROBABILITY DISTRIBUTION: Uniform distribution is a type of probability distribution in which all possible outcomes are equal. For instance, when a coin is tossed, the chance of getting a tail or head is the same.C. EXPONENTIAL PROBABILITY DISTRIBUTION: It is a continuous distribution that is commonly used to measure the expected time for an event to occur. It is often used to model the time elapsed between events. (2) r =6 N=26 N-r = 20 n =7, x=4, n-x=3 (3) Using formula for geometric distributionWhere x=n-1 and q= 1-pTherefore P = 0.35n =8 Therefore x= 8-1 =7 q=1-p=1-0.35=0.654.) When x=0 When x=7

POISSON DISTRIBUTION

Poisson Probability Distribution is another Discrete Random Variable. The Poisson Distribution is used to measure Time,Frequency,and Rate. In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

The Greek Letter Lamba(λ) is called the parameter of distribution which represents average number of times the Random Variable occurs in a specified interval or Space.

A discrete random variable is said to have a Poisson distribution, with parameter λ>0. It has a probability mass function given by:

P(X=x)=(λ^x × e^-λ)/x!

where:

x is the number of occurrences

e is Eulers number (e=2.7183)

! is the factorial function.

UNIFORM DISTRIBUTION

In probability theory and statistics, the uniform distribution also called rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)). Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution.The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable.

The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself (but it is dependent on the interval size), so long as the interval is contained in the distribution’s support.

It has a probability mass function is given by:

P(X=x)=1/b-a

EXPONENTIAL PROBABILITY DISTRIBUTION

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution the mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by. E(X)=1/λ,.

The exponential distribution is the only continuous probability distribution that has a constant failure rate.

P(X=x)= λe^-λx

2.

Six(6) doctors and 20 nurses attended a small conference,26 names are put In a hat, 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P (4 docs and 3 nurses): P(X=4D) × P(X=4N)

For P(X=4D)

r=6

n=7

N=26

x= 4 doctors

C = combination

D = Doctors

P(X=x)= [rCx] . [N-rCn-x]/NCn

= (6C4)(20C3)/26C7

=(15 × 1140)/657800

=17100/657800

=0.026.

For P(X=3N)

r=20; n=7; N=26 ; x= 3 Nurses; C = combination; N = Nurse

P(X=x)= [rCx] . [N-rCn-x]/NCn

= (20C3)(6C4)/26C7

=(15 × 1140)/657800

=17100/657800

=0.026.

Therefore, P (4 docs and 3 nurses) = 0.026 × 0.026

= 0.000676

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8 relationship.

Formula for geometric distribution; pq^x

Given

P=0.35, x= 8-1 = 7, q = 1- 0.35 = 0.65

P(x=8) = (0.35) (0.65)^7

=(0.35)(0.049)

P(x=8) = 0.0176.

4.

To make a dozen chocolate chip cookies, 50 chocolate chips are mixed into the dough, the same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lambda to the nearest whole number, what is the probability that it contains?

-No chocolate

-7 chocolate

Formula for Poisson probability distribution

P(X=x)=(λ^x × e^-λ)/x!

Given

λ = 50/12 = 4.16

λ = 4( to the nearest whole number)

e = 2.718

(a). P(x=0) ={2.718^⁻⁴ × 4^0}/0!

={0.0183 × 1}/1

=0.0183

b. P(x=7) = 2.718^⁻⁴ × 4^⁷ /7!

={0.0183 × 16384}/5040

=299.8/5040

=0.0594

A, Poisson probability distribution: measures how many times an event is likely to occur within ” x” period of time

B, uniform probability distribution: is a continuous probability distribution and is concerned with events that are equally likely to occur

C, exponential probability distribution: is often concerned with the amount of time until specific event occurs.

S2, Six doctors and twenty nurses attend a small Conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 Doctors and 3 Nurses are picked ?

N = 26 ( 6 + 20 =26)

n = 5, r = 6, x = ?

= = = = 0.0260.

3, Using the geometric, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

=

P = 0.35 1- P = 0.65 x = 7

0.35 X 0.0490 = 0.0172

4, To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. if a cookie is chosen at random from a large batch, also using lambda (the mean) to the nearest whole number. What is the probability that it contains:

a, No chocolate chips?

b, 7 chocolate chips?

Pq

Mean = to the nearest whole number 4

Mean = 4, x = 0

=

= 0.0183

B, 7 Chocolate Chips?

=

=

=

NAME: NWAIGBO NZUBECHUKWU VICTORY

REG NO: 2019/247274

DEPARTMENT: ECONOMICS DEPARTMENT

COURSE: ECONOMICS STATISTICS (EC0 231)

ASSIGNMENT

What short notes on the following types of Probability Distribution

Poisson Probability Distribution

A Poisson Probability Distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. It is a count distribution , Poisson distribution are often used to understand independent event that occur at a constant rate within a given interval of time.

Poisson distribution is a discrete distribution that measures the probability of a given number of event happening in a specified time period. Call centers uses Poisson distribution to model the number of expected calls per hour that they receive so they know how many call center reps to keep on staff.

Uniform Probability Distribution

A Uniform Probability Distribution refers to a type of probability distribution in which all outcome are equally likely. The Uniform Probability Distribution is a continuous probability distribution and is concerned with event that are equally likely to occur.

there are two types of uniform distribution which are the discrete and the continuous distribution.

Exponential Probability Distribution

The Exponential Probability Distribution is often concerned with the amount of time until some specific event occurs . It is the probability distribution of the time between events in a Poisson point process that is, a process in which event occur continuously and independently at a constant average rate for example, the amount of time (beginning now) until an earthquake occurs, length of time in minutes.

2, Six doctors and twenty nurses attend a small Conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 Doctors and 3 Nurses are picked ?

N = 26 ( 6 + 20 =26)

n = 5, r = 6, x = ?

= = = = 0.0260.

3, Using the geometric, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

=

P = 0.35 1- P = 0.65 x = 7

0.35 X 0.0490 = 0.0172

4, To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. if a cookie is chosen at random from a large batch, also using lambda (the mean) to the nearest whole number. What is the probability that it contains:

a, No chocolate chips?

b, 7 chocolate chips?

P q

Mean = to the nearest whole number 4

Mean = 4, x = 0

=

= 0.0183

B, 7 Chocolate Chips?

=

= P(X=7)

= 0.0595

1, a, The Poisson distribution is a discrete probability distribution we make use of when we are concerned with the number of times an event occurs within a specified period of time.

b, The Uniform distribution is a discrete probability distribution that is used to classify those kinds of probability distribution in which every event or possible outcomes all have equal likelihood of occurring.

c, The Exponential distribution is a continuous probability distribution that is used to model the time elapsed before a given event occurs.

2, This is solved using the hyper geometric distribution formula i.e. (rCx)*((N-r)C(n-x))/(NCr). The variables gotten are as follows: r=6, x=4, N=26, n=7.

After inputting the above variables into the hyper geometric distribution formula we arrive at “0.026”.

3, This is solved using the geometric distribution formula i.e. (p)(q^x). The variables gotten are as follows: p=0.35, q=0.65, x=7.

After inputting the above variables into the geometric distribution formula we arrive at “0.0172”.

4, a, This is solved using the Poisson distribution formula i.e. (((e^(-λ))(( λ^(x)))/(x!). The variables gotten are as follows: λ=50, x=0.

After inputting the above variables into the Poisson distribution formula we arrive at “(1.9287498×10)^(-22)”

b, This is also solved using the Poisson distribution formula i.e. (((e^(-λ))(( λ^(x)))/(x!). The variables gotten are as follows: λ=50, x=7.

After inputting the above variables into the Poisson distribution formula we arrive at “(3 × 10)^(-14)”

NWAFOR EMMANUEL ONYEDIKACHI; 2019/250914

EMMANUEL.NWAFOR.250914@UNN.EDU.NG

NAME: IDAJOR, JOHN AYUOCHIEYI

DEPARTMENT: ECONOMICS (MAJOR)

REG. NUMBER: 2019/248707

SOLUTIONS

(i) Poisson probability distribution: A poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after a French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

A Poisson distribution can be used to estimate how likely it is that something will happen “X” number of times. For example, if the average number of people who buy cheeseburgers from a fast-food chain on a Friday night at a single restaurant location is 200, a Poisson distribution can answer questions such as, “What is the probability that more than 300 people will buy burgers?” The application of the Poisson distribution thereby enables managers to introduce optimal scheduling systems that would not work with, say, a normal distribution.

b. Uniform probability distribution: The uniform probability distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. Put in another way, the uniform probability distribution is a type of probability distribution in which all outcomes are equally likely. There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5.

Some of the things to note about uniform probability are as follows: Uniform distributions are probability distributions with equally likely outcomes.

In a discrete uniform distribution, outcomes are discrete and have the same probability.

In a continuous uniform distribution, outcomes are continuous and infinite.

In a normal distribution, data around the mean occur more frequently.

The frequency of occurrence decreases the farther you are from the mean in a normal distribution

C. Exponential probability distribution: The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

To get this we use the hyper geometric distribution. Thus, P(X=4 doctors and 3 nurses) N= 6+20=26 n= 7 r= 6 (i.e. success) N-r= 19 (i.e. success failure =26 6)

Using the hyper geometric formula: P(X=x)=(xr)(N-rn-x) (Nn) = P(X=4 doctors and 3 nurses = (46)(319) (726)

=6C10 x 19C3 26C7

=15 x 969 657800 =0.0221

This means that the failure of a female celebrity to getting married will occur seven times before she finally gets married on the eight trial. Thus using the geometric formula below, we calculate.

P(X=x) = P(1-P)x or Pqx where x=0,1,2,

P= Probability of success on trial x= Number of failure q= Probability of event not occurring

Thus, our output is as follows: x=7 q= 0.65 (i.e. 1-0.35) P=0.35 Thus, P=(X=7) = 0.35 x (0.65)7 = 0.35 x 0.04902 = 0.0176

Given output: e= 2.71828 x=0 & 7 λ=? In order to find our Lamba (λ), we have to find the proportion of making a dozen of chocolate and approximate to the nearest whole number. Thus, λ= 50÷12 = 4.167 = 4 to the nearest whole number. λ= 4

Using the poisson’s formula when the probability that it contains o chips we have: e-λ λx x!

we have: (2.71828)-4 x (4)0 0!

= 0.01832 x 1 1 = 0.01832

The probability that it contain seven chips we have: e-λ λx x! (2.71828)-4 x (4)7 7! = 0.01832 x 16384 5040 = 300.15488 5040 =0.0596

Ugwuja Divine Uchenna

2019/244341

Diviboy123@gmail.com

Types of probability distribution

Exponential Distribution

the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. It is usually presented when we are dealing with events that are rapidly changing early on e.g news blog

The Uniform Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it, uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

Poisson Distribution

Poisson distribution, in statistics, a distribution function useful for characterizing events with very low probabilities of occurrence within some definite time or space. It is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution and is used to understand independent events that occur at a constant rate. It was named after French mathematician Simeon Denis.

It can be applied when:

You wish to count the number of times an event occurs in a given area of opportunity.

The number of events that occur in one area of opportunity is independent of the number of events that occur in other areas of opportunity

The probability that two or ore events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller

It can be mathematically represented as;

P(X=x) =

Where x = number of events in an area of opportunity.

Λ = expected number of events.

e = base of the natural logarithm system

mean µ = λ

variance = λ

standard deviation =

Using the hypergeometric distribution, find the probability that 4 doctors and 3 nurses are picked.

P(X=x) = = = = = 0.026

N= 26

n= 7

r= 6

x= 4

Using geometric distribution find out the probability that the female celebrity gives birth on the 8th try

P(X=x) = p* qx

P= 0.35 q=1-p= 0.65 x=7

P(X=7) = (0.35) (0.65)7

= (0.35) (0.0490)

=0.712

Using poisson distribution what is the probability that a cookie contains:

No chocolate chips

P(X=x) =

=

=

=0.0183

7 chocolate chips

P(X=7) =

=

==

=0.0595

Poisson Probability Distribution

This is a discrete distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if their events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

Poisson distribution is also used when you are interested in the number of times an event occurs in a given area of opportunity. Examples include: the number of scratches in a car’s paint, the number of mosquito bites on a person e.t.c.

Poisson Distribution Formula = P(x=x) =(e^(-⋋) ⋋^x)/X!

Uniform Probability Distribution

This is a continuous probability distribution and it is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

This means that the probability of each occurring is the same. As one of the simplest possible distributions, the uniform distribution is sometimes used as the null hypothesis, or initial hypothesis in hypothesis testing. The notation for the uniform distribution is X~U(a,b) when a =lowest value of X and b= the highest value of X. the probability density function is f(x) = Ib-a; for a≤x≤b

Exponential Probability Distribution

This is the probability distribution of the time between events in a poisson point process, i.e, a process in which the events occur continuously and independently at a constant average rate. It is a particularcase of the gamma distribution. This is also a continuous distribution that is commonly used to measure the expected time for an event to occur.

P (x = 4 doctors and 3 nurses)

N = 6 + 20 = 26

n = 7

r = 6

P (x=x) = ((■(r@x)) (■(N-r@n-x)))/((■(N@n)) )

= P (x = 4 doctors and 3 nurses) = (■(6@4))(■(20@3))/■(26@7)

=(15*1140)/657,800=0.02599

P (x), This implies that failure will occur 7 times before she got married ( at the 8th trial i.e success)

Let x be the number of her relationships before the marriage.

P (x = 7) = pqx where q = (1 – p)

P (1 – p)x

P = 0.35 q = 1 – 0.35 = 0.65

= 0.35 * (0.65)7

= 0.35 * 0.04902

= 0.017157

P(x = x) (e^(-⋋) ⋋^x)/X!

⋋ = 50/12 = 4.1666 ≈ 4

P (x = 0) = (e^(-⋋) ⋋^x)/X!

= (〖2.7183〗^(-4).4^0)/0!

= (1/54.5996*1)/1

= 0.01831

p (x = 0) 0.01831

P (x = 7) = (e^(-⋋) ⋋^x)/X!

= (〖2.7183〗^(-4).4^7)/7!

= (0.01831*.16384)/5040

= 0.05952

p (x = 7) = 0.05952

ASSIGNMENT

1. Write short notes on the following types of probability distributions

A] Poisson probability distribution

This is a type of discrete probability distributions that is used to show how many times an event is likely to occur over a specified period. Therefore it is a count distribution. It was named after a French mathematician Simeon Denis Poisson. It variable can only take a specific value in an infinite list e.g. the values 0,1,2,3 etc with no fractions or decimals. It is used to measure time and frequency.

Formulae: P(X=x) = e-✓\ X ✓\x x!

Where; e is exponential or base. It is a constant (2.7183). √ is a lambda, the expected number of events. X is the number of event in an area.

The mean = ✓\ = variance. Standard deviation = ✓\1/2

b] Uniform probability distribution

This refers to a continuous type of distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, club, spade, and or diamond is equally likely. The uniform distribution can be visualized as a straight horizontal line. So for a coin flip returning a head or a tail, both have a probability of p = 0.50 and would be represented as a line from the y-axis at 0.50.

Formulae: the notation for the uniform distribution is X ~U (a, b) where a = the lowest value of x and b = the highest value of x

The probability density function is

F(x) = 1

——- For a b

B – a

c] Exponential probability distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process i.e. a process in which events occurs continuously and independently is a constant average rate. It is the continuous analogue of the geometric distribution, and it has the character of being memory less, it can also be found in various contexts.

Formula: f(x; ✓\) = (✓\e-✓\x x>0

(0 x 0 is the parameter of the distribution often called lambda or the rate parameter. The distribution is supported on the interval [0, ~). If a random variable X has this distribution, we write X ~ Exp(✓\).

Mean = 1/✓\ variance = 1/✓\2 standard deviation = (1/✓\2)1/2

The exponential distribution exhibit infinite divisibility.

2] Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

P(X=x) = (r) (N-r) where; r is 6, x is 4, N is 26, n is 7

(x) (n-x)

———————

(N)

(n)

P(X=4 doctors and 3 nurses) = (6) (26-6)

(4) (7-4) = 6C4 X 20C3

—————————- ————————–

(26) 26C7

(7)

P(X=4 doctors and 3 nurses) = 15 X1140 17100

———————– = ———–

657800 657800

P(X=4 doctors and 3 nurses) = 0.025996

————

3] Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship?

Formulae: pqx Or P (1-p)x. Where p is 0.35, q or 1- p is 0.65 and X is 7 (number of failures before success).

P(X=7) = 0.35X0.657 = 0.35X0.0492

P(X=7) = 0.0172

4} to make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using Lambda (the mean) to the nearest whole number. What is the probability that it contains:

Solve by Poisson probability distribution

A] No chocolate chips?

Formulae: P(X=x) = e-✓ X √x = 500 X e-50 = 1 X (192.874984796)*10-24 x! 0! 1

P(X=0) = (192.874984796)*10-24

B} 7 chocolate chips?

Formulae: P(X=x) = e-✓ * √x = 507 X e-50 x! 7!

= 781250000000X (192.874984796)*10-24 5040

P(X=7) = (298.975360858)*10-16

NAME: Dike John Chukwudozie

Reg no; 2018/241837

1. Write short notes on:

*Poisson Probability Distribution*

The Poisson distribution is a discrete probability function that means the variable can only take specific values in a given list of numbers, probably infinite. A Poisson distribution measures how many times an event is likely to occur within “x” period of time. In other words, we can define it as the probability distribution that results from the Poisson experiment. A Poisson experiment is a statistical experiment that classifies the experiment into two categories, such as success or failure. Poisson distribution is a limiting process of the binomial distribution.

*Uniform Probability Distribution*

The Uniform distribution is the simplest probability distribution, but it plays an important role in statistics since it is very useful in modeling random variables. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur.

*Exponential Probability Distribution*

The exponential distribution is a right-skewed continuous probability distribution that models variables in which small values occur more frequently than higher values. Small values have relatively high probabilities, which consistently decline as data values increase. Statisticians use the exponential distribution to model the amount of change in people’s pockets, the length of phone calls, and sales totals for customers. In all these cases, small values are more likely than larger values.

Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

Name: Appolos sopuruchukwu bethel

Reg No: 2019/244006

Email: Appolosbethel2019@gmail.com

Exponential Distribution

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. We will now mathematically define the exponential distribution, and derive its mean and expected value. Then we will develop the intuition for the distribution and discuss several interesting properties that it has.

An interesting property of the exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution. To see this, recall the random experiment behind the geometric distribution: you toss a coin (repeat a Bernoulli experiment) until you observe the first heads (success). Now, suppose that the coin tosses are Δ seconds apart and in each toss the probability of success is p=Δλ. Also suppose that Δ is very small, so the coin tosses are very close together in time and the probability of success in each trial is very low. Let X be the time you observe the first success. We will show in the Solved Problems section that the distribution of X converges to Exponential (λ) as Δ approaches zero.

To get some intuition for this interpretation of the exponential distribution, suppose you are waiting for an event to happen. For example, you are at a store and are waiting for the next customer. In each millisecond, the probability that a new customer enters the store is very small. You can imagine that, in each millisecond, a coin (with a very small P(H)) is tossed, and if it lands heads a new customers enters. If you toss a coin every millisecond, the time until a new customer arrives approximately follows an exponential distribution.

The above interpretation of the exponential is useful in better understanding the properties of the exponential distribution. The most important of these properties is that the exponential distribution has less memory. To see this, think of an exponential random variable in the sense of tossing a lot of coins until observing the first heads. If we toss the coin several times and do not observe a heads, from now on it is like we start all over again. In other words, the failed coin tosses do not impact the distribution of waiting time from now on. The reason for this is that the coin tosses are independent. We can state this formally as follows:

P(X>x+a|X>a)=P(X>x).

Uniform Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a

heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

POINTS

Uniform distributions are probability distributions with equally likely outcomes.

In a discrete uniform distribution, outcomes are discrete and have the same probability.

In a continuous uniform distribution, outcomes are continuous and infinite.

In a normal distribution, data around the mean occur more frequently.

The frequency of occurrence decreases the farther you are from the mean in a normal distribution.

Understanding Uniform Distribution

There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. There are only 6 possible values to return and nothing in between.

The plotted results from rolling a single die will be discretely uniform, whereas the plotted results (averages) from rolling two or more dice will be normally distributed.

Some uniform distributions are continuous rather than discrete. An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there is an infinite number of points between 0.0 and 1.0.

There are several other important continuous distributions, such as the normal distribution, chi-square, and Student’s t-distribution.

There are also several data generating or data analyzing functions associated with distributions to help understand the variables and their variance within a data set. These functions include probability density function, cumulative density, and moment generating functions.

Poisson Distribution

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

POINTS

A Poisson distribution, named after French mathematician Siméon Denis Poisson, can be used to estimate how many times an event is likely to occur within “X” periods of time.

Poisson distributions are used when the variable of interest is a discrete count variable.

Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

Understanding Poisson Distributions

A Poisson distribution can be used to estimate how likely it is that something will happen “X” number of times. For example, if the average number of people who buy cheeseburgers from a fast-food chain on a Friday night at a single restaurant location is 200, a Poisson distribution can answer questions such as, “What is the probability that more than 300 people will buy burgers?” The application of the Poisson distribution thereby enables managers to introduce optimal scheduling systems that would not work with, say, a normal distribution.

One of the most famous historical, practical uses of the Poisson distribution was estimating the annual number of Prussian cavalry soldiers killed due to horse-kicks. Modern examples include estimating the number of car crashes in a city of a given size; in physiology, this distribution is often used to calculate the probabilistic frequencies of different types of neurotransmitter secretions. Or, if a video store averaged 400 customers every Friday night, what would have been the probability that 600 customers would come in on any given Friday night?

Solution

2) 0.0260

==

3) P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

4) 50/12 = 25/6 = 4.166 (to the nearest whole number = 4)

a) P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

a) P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

ASSIGNMENT ON ECO 231 Date: 24/11/2021

Answers to question 1: (I) Poisson Probability Distribution?

Poisson distribution is used when one is interested in the number of times an event occurs in a given area of opportunity.Poisson probability distribution is a discrete variable. Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

USES OF POISSON PROBABILITY DISTRIBUTION

1)Poisson distributions are used by businessmen to make forecasts about the number of customers or sales on certain days or seasons of the year.

2)With the Poisson distribution, companies can adjust supply to demand in order to keep their business earning good profit. In addition, waste of resources is prevented.

3)By using this tool, businessmen are able to estimate the time when demand is unusually higher, so they can purchase more stock.

(II) The Uniform Probability Distribution

This is usually present when we are dealing with events that are rapidly changing early on. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

Uniform distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. The probability is constant since each variable has equal chances of being the outcome.

Example of Uniform Distribution

If you stand on a street corner and start to randomly hand a $100 bill to any lucky person who walked by, then every passerby would have an equal chance of being handed the money. The percentage of the probability is 1 divided by the total number of outcomes (number of passersby). However, if you favored short people or women, they would have a higher chance of being given the $100 bill than the other passersby. It would not be described as uniform probability.

Types of Uniform Distribution

1) Discrete uniform distribution

In statistics and probability theory, a discrete uniform distribution is a statistical distribution where the probability of outcomes is equally likely and with finite values.

2)Continuous uniform distribution

Not all uniform distributions are discrete; some are continuous. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. Unlike discrete random variables, a continuous random variable can take any real value within a specified range.A continuous uniform distribution usually comes in a rectangular shape. A good example of a continuous uniform distribution is an idealized random number generator. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. However, there is an infinite number of points that can exist.

Mean (A + B)/2

Median (A + B)/2

Range B – A

Standard Deviation (B−A)212−−−−−√

Coefficient of Variation (B−A)3√(B+A)

Skewness 0

Kurtosis 9/5

(III) Exponential Probability distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.The exponential distribution refers to the continuous and constant probability distribution which is actually used to model the time period that a person needs to wait before the given event happens and this distribution is a continuous counterpart of a geometric distribution that is instead distinct.

A continuous random variable X is said to have an exponential distribution with parameter λ>0, shown as X∼Exponential(λ), if its PDF is given by

fX(x)={λe−λx0x>0otherwise.

Answer to question 2:

N=26,n=7,x=(4 doctors and 3 nurses),r=6

P(X=x)= (6C4)(20C3)/(26C7)=0.0259

Answer to question 3:

P(X=x)= p(1-p)*

0.35(0.65)^7=0.01716

Answer to question 4a: Using poisson formula, we get 0.00000000000000000000019288

4b) we get 0.00000000000002989

NAME: UGWOKE MICHAEL-MARY IKECHUKWU

DEPARTMENT: ECONOMICS

REGISTRATION NUMBER: 2019/248716

1a.) Poisson Probability Distribution: Poisson Probability distribution shows how many times an event is likely to occur over a specific period of time. It helps to predict the probability of an event happening when you know how often the e vent has occur.

These was conceptualized by French mathematician, Simeon Denis Poisson

1b.) Uniform Probability Distribution: Uniform distribution is type of probability distribution in which all the outcomes are equally likely.

For instance, when a coin is tossed, the chance of getting a tail or head is the same

1c.) Exponential Probability Distribution: This is a continuous distribution that is commonly used to measure the expected time for a event to occur. It is often used to model the time elapsed between events

2.)

r =6 N=26

N-r = 20

n =7, x=4, n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.) Using formula for geometric distributionP(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261≈0.05954

Name: ugwunze Emmanuel chukwuebuka

Reg No: 2019/245483

Email: ugwunzeemmanuel@gmail.com

Exponential Distribution

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. We will now mathematically define the exponential distribution, and derive its mean and expected value.

Uniform Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a

heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

Poisson Distribution

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

Solution

==

3) P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

4) 50/12 = 25/6 = 4.166 (to the nearest whole number = 4)

P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

1a. Poisson distribution is one that measures the probability of a given number of events happening in a specified time period. In finance, the

b. The exponential distribution is a continuous distribution that is commonly used to measure the expected time for an event to occur.

c. A uniform distribution, also called a rectangular distribution, is a probability distribution that has constant probability.

2. (r¦x)((N-r)¦(n-x))

(█(N@n))

(█(6@4))(█(20@3))

(26¦7) =

3. P (X=x)= P(1-P)x

Where p = 0.35, `1-p = 1-0.35= 0.65

P(X-x)= 0.35 ᵡ 0.657 = 0.0004182

4. P (X-x)= ℮-λ λx

X!

Where ℮= 2.71828

λ= 50∕12≈4

x=0

P(X-x)= 2.71828-4∙40 = 0.01832

0!

Where x=7

P (X-x)= 2.71828-4∙47 = 0.23822

7!

1. (a) Poisson distribution: this is a type of discrete probability distribution that is used to determine the number of times an event occurs in a given area of opportunity. There is unlimited number of outcomes unlike binomial distribution. Example, number of scratches in a car’s paint.

(b) Uniform probability distribution: this is a type of uniform distribution in which all outcomes are equally likely. Example, a deck of cards.

© Exponential probability distribution: this is a type of discrete probabity distribution that is usually present when we are dealing with events that are rapidly changing. Example, news, blog, etc.

2. = = 0.025996

3. P(X=7) = pq7 = (0.35) (0.65)7 = 0.017158

4. (℮^-ƛ) (ƛx)/xǃ =

a) 0.000006144 b) 0.00000045967

a Poisson probability distributions:The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time.

b. Uniform probability:

c. Exponential probability distribution:

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

= 0.0259

x=3

6 20

3 4

26

7

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

= 0.018

P(x=7)

=0.059

Okafor Roseline Chugo

2019/248202

Economics

Eco 231 Assignment

(1)

Write short notes on the following types

of probability distribution:

a. Poisson probability distribution

b. Uniform probability distribution

c. Exponential probability distribution

Poisson probability distribution:

Poisson distribution is a probability

distribution that is used to

show how many times an event is likely

to occur over a specified period. In

other words, it is a count

distribution. Poisson distributions are often used to understand independent

events that occur at a constant rate

within a given interval of time.

The Poisson distribution is a discrete

function, meaning that the variable

can only take specific

values in a (potentially infinite) list.

Put differently, the variable cannot

take all values in any

continuous range. For the Poisson

distribution (a discrete distribution), the

variable can only take the

values 0, 1, 2, 3, etc., with no fractions

or decimals.

The Formula for the Poisson

Distribution given as: P(X=x)=

�−λ .λ � Where:

e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of

x when that is also equal to its variance

Uniform Probability Distribution:

The uniform distribution is a continuous

probability distribution

and is concerned with events that are

equally likely to occur. When working

out problems that have a uniform

distribution, be careful to note if the

data is inclusive or exType equation

here.clusive.

It is also a term used to describe a form

of probability distribution where every

possible outcome has an equal likelihood of happening. The

probability is constant since each

variable has equal chances of being the

outcome.

The notation for the uniform

distribution is X ~ U (a, b)

where a = the lowest value of x and b =

the highest value of x.

There are two types of uniform

distribution:

➢ Discrete uniform distribution

➢ Continuous uniform distribution

Discrete uniform distributions: These

have a finite number of outcomes.

Continuous uniform distribution:

This is a statistical distribution with

an infinite number of

equally likely measurable values.

Exponential probability

distribution: The exponential

distribution is often concerned with the

amount of time until some specific

event occurs. For example, the amount

of time (beginning now) until an earthquake occurs has an

exponential distribution. Other

examples include the length of time,

in minutes, of long-distance business

telephone calls, and the amount of

time, in months, a car

battery lasts. Exponential distributions

are commonly used in calculations of

product reliability, or

the length of time a product last.

The formula for the exponential

distribution: P(X=x) =me-mx=1μe-1μx

Where m = the rate parameter, or μ =

average time between occurrences.

(2)

Six doctors and twenty nurses attend a

small Conference. All 26 names are put

in a hat, and 7 names

are randomly selected without

replacement. What is the probability

that 4 Doctors and 3 Nurses are

picked?

�(�=�)(��)(�−��−�)

(��)

N = 26 (6 + 20 =26)

n = 5, r = 6, x = ?

�(�=4 ������� ��� 3 ������) (64

)(25−67−4)(267)

(64)(203)(267) = 6�4 � 20�

326�7 = 15 � 1140657800 17100657800 = 0.0260.

(3)

Using the geometric, suppose that the

probability of female celebrity getting

married is 0.35. Find the probability that

she will actually get married in her 8th relationship. = �( 1−�)�

P = 0.35 1- P = 0.65 x = 7

�(�= 7) =0.35 (0.65)7

0.35 X 0.0490 = 0.0172

(4)

To make a dozen chocolate cookies,

50 chocolate chips are mixed into

the dough. The same proportion is

used for all batches. if a cookie is

chosen at random from a large batch,

also using lambda (the mean) to the

nearest whole number. What is the

probability that it contains?

a, No chocolate chips?

b, 7 chocolate chips?

�(�=�) Pq

Mean = 5012=4.17 to the nearest whole

number 4

Mean = 4, x = 0

= �(�=0) = 2.7183−4 ∗ 400!

=

0.0183∗1

1 = 0.0183

B. 7 Chocolate Chips?

= �(�=7) = 2.7183−4∗ 477! = �(�=7)

= 0.0183∗163845040

= �(�=7)=0.0595

a Poisson probability distributions:

Poisson distribution is a probability that is used to show how many times an event occurs over a specified period. The Poisson distribution is a discreet function, meaning that the variable can only take one specific value in a list.

b. Uniform probability:

Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely.

c. Exponential probability distribution:

This is a probability distribution of the time between events in a point process that is a process in which events occur continuously and independently at a constant average rate.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

=0.0259

x=3

6 20

3 4

26

7

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

= 0.018

P(x=7)

=0.059

a Poisson probability distributions:

Probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

b. Uniform probability:

uniform distribution refers to a type of probability distribution in which all outcomes are equally likely

c. Exponential probability distribution:

exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

= 0.0259

x=3

6 20

3 4

26

7

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

= 0.018

P(x=7)

=0.059

1a Poisson Distribution

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The Poisson Distribution formula is: P(x; μ) = (e-μ) (μx) / x!

Where,

e is the base of the logarithm

x is a Poisson random variable

λ is an average rate of value

1b. A uniform distribution, also called a rectangular distribution, is a probability distribution that has constant probability. The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur.

This distribution is defined by two parameters, a and b:

a is the minimum.

b is the maximum.

The distribution is written as U(a, b).

1c. Probability theory and statistics, the exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens. It is a process in which events happen continuously and independently at a constant average rate. The exponential distribution has the key property of being memoryless. The exponential random variable can be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the supermarket follows an exponential distribution.

Exponential Distribution Formula

Exponential distribution formula. The main formulas used for analysis of exponential distribution let you find the probability of time between two events being lower or higher than x: P(x>X) = exp(-a*x) P(x≤X) = 1 – exp(-a*x)

2. P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

3. a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

Ok. Okoro David KOSISOCHUKWU 2019/241946

a Poisson probability distributions:

Poisson distribution is a probability that is used to show how many times an event occurs over a specified period. The Poisson distribution is a discreet function, meaning that the variable can only take one specific value in a list.

b. Uniform probability:

Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely.

c. Exponential probability distribution:

This is a probability distribution of the time between events in a point process that is a process in which events occur continuously and independently at a constant average rate.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

=(6C4 × 20C3)/26C7

=17100/657800

x=3

6 20

3 4

26

7

=(6C3×20C4)/26C7

=96900/657800

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 × 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

=(e^(-4)×4^0)/0!

= 0.018

P(x=7)

=(e^(-4)×4^7)/7!

=(0.018×16384)/5040

=0.059

1a) Poisson distribution: Any of a class of discrete probability distributions that express the probability of a given number of events occurring in a fixed time interval, where the events occur independently and at a constant average rate.

Examples : the number of injuries in a working place, the number of things sold at a time in a shop.

1b) Uniform probability distribution: this is a symmetric probability distribution wherein every outcome is equally likely to occur at any point in the distribution.

There are two types of uniform probability distribution ; discrete and continuous. Examples are deck of cards and coins.

1c) Any of a class of continuous probability distributions used to model the time between events that occur independently at a constant average rate.

2) hypergeometric distribution

P(X=x) (rCc) (N-r C n-x) / (NCn)

Where N=26

Where n=7

Where r=6

Where c= combination

(6c4)(26-6 C 7-4) all divided by (26 C 7)

= 15×1140 all divided by 657800

= 0.0259

3) geometric distribution = pq^x

Where p = 0.35

Where q= (1-p) = 1-0.35 = 0.65

Where x=8-1, x=7

pq^x= 0.35×0.65^7

= 0.0172

4a) using poisson distribution

P(X=x)

e= lambda which is 2.7183

∆= mean which is 1

x= 0

(e^-∆)(∆^x) all divided by x!

=(2.7183^-1)(1^0) all divided by 0!

= 0.3679 ×1 all divided by 1

= 0.3679

4b) where x= 7

= (2.7183^-1)(1^7) all divided by 7!

= 0.3679 ×1 all divided by 5040

= 0.000072

= 0.00001

Name: NNA OZIOMA VINE

Reg No: 2019/247263

Email: nnaozioma71@gmail.com

1. Exponential Distribution

This is a probability of time between events in a Poisson process. A process in which events occur continously and independently at a constant average rate it has the memory less. We can state this formally as follows:

P(X>x+a|X>a)=P(X>x).

2. Uniform Distribution

This refers to a probability in which outcomes are equally a like. There are 2 kinds of probability they are: Discreet probability and Continous probability. Example is a probability of a coin when a coin is tossed the probability of getting either a head or tail is the same.

4. POISSION PROBABILITY DISTRIBUTION

This is a probability distribution that is used to show how many times an event is likely to happen in a give period. The Poisson distribution is a discrete function meaning that the variable can only take specified season in a list. It was named after a French mathematian SIMEON DENIS. Values like 0,1,2,3 or with no fractions or decimals.

The Formula for the Poisson distribution is

Solution

n formula

==

P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

50/12 = 25/6 = 4.166 (to the nearest whole number = 4)

P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

Name: oguaju Adaobi Dorathy

Reg No: 2016/234292

Email: nnaozioma71@gmail.com

1. Exponential Distribution

This is a probability of time between events in a Poisson process. A process in which events occur continously and independently at a constant average rate it has the memory less. We can state this formally as follows:

P(X>x+a|X>a)=P(X>x).

2. Uniform Distribution

This refers to a probability in which outcomes are equally a like. There are 2 kinds of probability they are: Discreet probability and Continous probability. Example is a probability of a coin when a coin is tossed the probability of getting either a head or tail is the same.

4. POISSION PROBABILITY DISTRIBUTION

This is a probability distribution that is used to show how many times an event is likely to happen in a give period. The Poisson distribution is a discrete function meaning that the variable can only take specified season in a list. It was named after a French mathematian SIMEON DENIS. Values like 0,1,2,3 or with no fractions or decimals.

The Formula for the Poisson distribution is

Solution

n formula

==

P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

50/12 = 25/6 = 4.166 (to the nearest whole number = 4)

P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

Poisson probability distributions:

Poisson distribution is a probability that is used to show how many times an event occurs over a specified period.

b. Uniform probability:

Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely.

c. Exponential probability distribution:

This is a probability distribution of the time between events in a point process that is a process in which events occur continuously and independently at a constant average rate.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

=0.0259

x=3

6 20

3 4

26

7

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

= 0.018

P(x=7)

=0.059

Name: Chisalum Emmanuel Chinecherem

Reg No: 2019/249408

Email: nnaozioma71@gmail.com

1. Exponential Distribution

This is a probability of time between events in a Poisson process. A process in which events occur continously and independently at a constant average rate it has the memory less. We can state this formally as follows:

P(X>x+a|X>a)=P(X>x).

2. Uniform Distribution

This refers to a probability in which outcomes are equally a like. There are 2 kinds of probability they are: Discreet probability and Continous probability. Example is a probability of a coin when a coin is tossed the probability of getting either a head or tail is the same.

4. POISSION PROBABILITY DISTRIBUTION

This is a probability distribution that is used to show how many times an event is likely to happen in a give period. The Poisson distribution is a discrete function meaning that the variable can only take specified season in a list. It was named after a French mathematian SIMEON DENIS. Values like 0,1,2,3 or with no fractions or decimals.

The Formula for the Poisson distribution is

Solution

n formula

==

P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

50/12 = 25/6 = 4.166 (to the nearest whole number = 4)

P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

What is poisson probability distribution: it is used when you are interested in the number of times an event occurs in a given area of opportunity. The area of opportunity is a continuous unit or interval of time such area in which more than one occurrence of an event occurs.

Examples

Number of scratches in a car’s paint

Number of mosquito bites on a person.

Uniform probability distribution:

Exponential probability distribution: is usually present when we are dealing with events that are rapidly changing. E.g news blog

2

2 Six(6) doctors and twenty(20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

P( x=4 doctors & 3 nurses)

N= 6+20=26

n= 7

r=6, p(x=×)= [rC×]. [N-rCn-×] ÷NCn

×= 4 ( 6C4)(20C3)÷ (26C7) = 6C4!(6-4)!* 20C3!(20-3)!÷ 26C7!(26-7)!= 0.02599574

2ii when ×=3 = p(X=×)= [rC×] [N-rn-×]÷ NCn

3] Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

Solution

Formula= p(X=×)= p(1-p)× or pq× = 0.35×8= 2.8

×=8

4.

44} To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using Lambda (the mean) to the nearest whole number. What is the probability that it contains:

a] No chocolate chips

b} 7 chocolate chips?

Solution

Formula

P[X=×]=e^-π*π^×÷x!

π=50, x=0, e= 2.7183

P[x=0]= 2.718350500÷0=0.000000000000000000192.815÷1= 000000000000192.815

4ii ×=7

P[X=×]= 2.7183-50507÷7!= 2.7183^-50×78150000000÷5040= 0.000000000000298.875

Name: Nwodo Micheal Chidera

Reg No: 2019/243281

Email: nnaozioma71@gmail.com

P(X>x+a|X>a)=P(X>x).

2. Uniform Distribution

The Formula for the Poisson distribution is

Solution

n formula

==

P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

OGAEME ONYEDIKACHI LOVEDEY

REG NO: 2019/251299

1a. Poisson probability distribution is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution and are often used to understand independent events that occur at a constant rate within a given interval of time. It is also a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals. The Formula for the Poisson Distribution Is

f(x) = P(X=x) = (e-λ λx )/x!

Where:

• e is Euler’s number (e = 2.71828…)

• x is the number of occurrences

• x! is the factorial of x

• λ is equal to an average rate of the expected value (EV) of x when that is also equal to its variance

The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity.

1b.Uniform probability Distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. The probability is constant since each variable has equal chances of being the outcome. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. It usually comes in a rectangular shape.The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

1c.Exponential probability distribution

The exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events.

2. P(X=4 doctors and 3 nurses), N=26, n=7, r=6

Using hyper geometric formula,

P(X=4 doctors 3 nurses)= [6C4 × (26-6)C(7-4)] ÷ (26C7)

=[15×1140]÷ (657800)

= 0.025995743

3. P(X=x)= p(1-p)^x or pq^x

Using geometric approach,

Where, p= 0.35 and x= 8-1=7

Let x be the number of relationship before the marriage

P(X=7)= 0.35×(1-0.35)^7

=0.35×(0.65)^7

=0.35×0.049022278

=0.017157797

4. f(x) = P(X=x) = (e^-λ* λ^x )/x!

Using poisson formula,

Where, e=2.7183, λ= 50÷12= 4.167and x= number of occurrence

a. P(X=0)= [(4.167^0)× 2.7183^(-4.167)] ÷ 0!

= 0.015498254

b. P(X=7)= [(4.167^7)×(2.7183^-4.167)]÷ 7!

= [(21815.46644)×(0.015498254)] ÷ 5040

=338.1016571÷5040

=0.067

a Poisson probability distributions: Probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

b. Uniform probability:uniform distribution refers to a type of probability distribution in which all outcomes are equally likely

c. Exponential probability distribution:exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

= 0.0259

x=3

6 20

3 4

26

7

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

= 0.018

P(x=7)

=0.059

1a) Poisson distribution: Any of a class of discrete probability distributions that express the probability of a given number of events occurring in a fixed time interval, where the events occur independently and at a constant average rate.

Examples : the number of injuries in a working place, the number of things sold at a time in a shop.

1b) Uniform probability distribution: this is a symmetric probability distribution wherein every outcome is equally likely to occur at any point in the distribution.

There are two types of uniform probability distribution ; discrete and continuous. Examples are deck of cards and coins.

1c) Any of a class of continuous probability distributions used to model the time between events that occur independently at a constant average rate.

2) hypergeometric distribution

P(X=x) (rCc) (N-r C n-x) / (NCn)

Where N=26

Where n=7

Where r=6

Where c= combination

(6c4)(26-6 C 7-4) all divided by (26 C 7)

= 15×1140 all divided by 657800

= 0.0259

3) geometric distribution = pq^x

Where p = 0.35

Where q= (1-p) = 1-0.35 = 0.65

Where x=8-1, x=7

pq^x= 0.35×0.65^7

= 0.0172

4a) using poisson distribution

P(X=x)

e= 2.7183

∆= mean or lambda which is 50/12, which is =4 (i.e to the nearest whole number)

x= 0

(e^-∆)(∆^x) all divided by x!

=(2.7183^-4)(4^0) all divided by 0!

= 0.0183×1 all divided by 1

= 0.0183

4b) where x= 7

= (2.7183^-4)(4^7) all divided by 7!

= 0.0183×16,384 all divided by 5040

= 0.0595

Ugwu kaosisochukwu immaculeta

2019/241226

kaosisochukwu.ugwu.241226@unn.edu.ng

1a. Poisson probability distribution is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution and are often used to understand independent events that occur at a constant rate within a given interval of time. It is also a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals. The Formula for the Poisson Distribution Is

f(x) = P(X=x) = (e-λ λx )/x!

Where:

• e is Euler’s number (e = 2.71828…)

• x is the number of occurrences

• x! is the factorial of x

• λ is equal to an average rate of the expected value (EV) of x when that is also equal to its variance

The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity.

1b.Uniform probability Distribution is a term used to describe a form of probability distribution where every possible outcome has an equal likelihood of happening. The probability is constant since each variable has equal chances of being the outcome. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. It usually comes in a rectangular shape.The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

1c.Exponential probability distribution

The exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events.

2. P(X=4 doctors and 3 nurses), N=26, n=7, r=6

Using hyper geometric formula,

P(X=4 doctors 3 nurses)= [6C4 × (26-6)C(7-4)] ÷ (26C7)

=[15×1140]÷ (657800)

= 0.025995743

3. P(X=x)= p(1-p)^x or pq^x

Using geometric approach,

Where, p= 0.35 and x= 8-1=7

Let x be the number of relationship before the marriage

P(X=7)= 0.35×(1-0.35)^7

=0.35×(0.65)^7

=0.35×0.049022278

=0.017157797

4. f(x) = P(X=x) = (e^-λ* λ^x )/x!

Using poisson formula,

Where, e=2.7183, λ= 50÷12= 4.167and x= number of occurrence

a. P(X=0)= [(4.167^0)× 2.7183^(-4.167)] ÷ 0!

= 0.015498254

b. P(X=7)= [(4.167^7)×(2.7183^-4.167)]÷ 7!

= [(21815.46644)×(0.015498254)] ÷ 5040

=338.1016571÷5040

=0.067

Name: Nwodo Micheal Chidera

Reg No: 2019/243281

Email: nnaozioma71@gmail.com

1. Exponential DistributionThis is a probability of time between events in a Poisson process. A process in which events occur continously and independently at a constant average rate it has the memory less. We can state this formally as follows:

P(X>x+a|X>a)=P(X>x).

2. Uniform Distribution

This refers to a probability in which outcomes are equally a like. There are 2 kinds of probability they are: Discreet probability and Continous probability.

Solution

n formula

P(X=7)

= (0.35)(0.65)7

= (0.35)(0.0490)

= 0.0172

P(X=0); λ=4

= e–4*40/0!

= 0.0183*1/1

= 0.0183

P(x=7)

= e–4*47/7!

= 0.0182*1.6384/5040

= 299.8272/5040

= 0.0595

a Poisson probability distributions: A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.

b. Uniform probability:uniform distribution refers to a type of probability distribution in which all outcomes are equally likely

c. Exponential probability distribution:exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

= 0.0259

x=3

6 20

3 4

26

7

=0.14731

p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 0.0490

=0.01715

e-ƛ × ƛx

x!

P(x=0)

= 0.018

P(x=7)

=0.059

NAME: Uzochukwu Chidinma Vivian.

Reg no: 2017/250786.

Write short notes on the following:-

A. Poisson probability Distribution is a discrete frequency distribution which gives the probability of a number of independent events occuring at a fixed time. Poisson distribution unlike binomial has to do with rate, frequency and time. It is one of the distributions for discrete variables.

A poisson distribution can also be referred to as a tool that helps to predict the probability of certain events happening when you know how often the event has occured.It gives the probability of a given number of events happening in a fixed interval of time. It is used to describe the distribution of rare events in a large population.

For Example:- In a cafe, the customer arrives at a mean rate of 2 per min. What is the probability of arrival of 5 customers in one minute using the poisson distribution.

B. Uniform probability Distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within its uniform distributions because the likelihood of drawing a heart,a club,a diamond or a spade is equally likely.In probability theory or statistics,the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

C. Exponential probability Distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. An interesting property of the exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution.

2. Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution:-

P(X=x)= r N-r x n-x N n

N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6 4 7-4 26 7

P(X=4) = 6C4 X 20C3 ÷ 26C7

P(X=4) = 15X1140÷657800

P(X=4) = 17100÷657800= 0.0025995743= 0.0260. 4dp3.

2. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

Solution:-

P(X=x)=Pqx P= 0.35q= 1-p, 1-0.35= 0.65 x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7= 0.017157=0.017157×100= 1.7% chance.4.

To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

a) No chocolate chips:-

Solution:-

P(X=x) = e-ƛƛx/x!ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183.

b) 7 chocolate chips:-

Solution:-

P(X=x) = e-ƛƛx/x!ƛ = 50/12 = 4 (to the nearest whole number)P(X=7) = e-4(4)7/7! = 0.0595.

NKEONYE OLUCHI PRAISE

2019/250120

1a) Poisson probability distribution: This is a probability distribution that is used to show how many times an even occurs over a specified period. In other words, it is a count distribution. It was named after French mathematician Siméon Denis Poisson. The poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list i.e the variable can only take the values 0,1,2,3,4 e.t.c with no fractions or decimals.

1b) Uniform probability distribution: Uniform probability distribution is a term used to describe a form of probability distribution where every possible outcomes has an equal likelihood of happening. The probability is constant since each variable has equal chances of being the outcome. This is the simplest statistical distribution.

1c) Exponential Probability Distribution: This is one if the widely used continuous probability distributions. It is often user to model the time elapsed between events. It is often used to answer in probabilistic terms questions such as:

– How much time will elapse before a tornado occurs in a given region?

– How long do we need to be wait until a customer enters our shop?

It has a memoryless property, which says that future probabilities do not depend on any past information.

2.) N=26

r=6

N-r= 20

n=7

x=4

n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.)

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

Reg No: 2016/233923

1. POISSON PROBABILITY DISTRIBUTION

(a.) Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

Poisson distribution formula: P(X=x)= e^-λ. λ^x / x!

(b.) The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. The data used in uniform distribution can be inclusive or exclusive. In a continuous discrete uniform probability distribution, outcomes are continuous and infinite. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

Uniform distribution formula: F(x)= 1/ b-a

(c.) Exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens. It is a process in which events happen continuously and independently at a constant average rate. The exponential distribution has the key property of being memoryless. The exponential random variable can be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the supermarket follows an exponential distribution.

Exponential distribution formula: F(x)= λe ^-λx

2. Hypergeometric probability distribution formula: P(X=x)= (r/x/) (N-r/n-x) / (N/x)

N = 26

n = 7

r = 6

X = 7

= (6/4) (20/3) / (26/7)

=6C4 × 20C3 / 26C7

15×1140 / 657800

= 0.0259 or 0.026

3. Geometric distribution formula: P(X=x) = pq^x

P= 0.35

q= 1- 0.35

X= 7

=0.35 × (0.65)^7

0.35 × 0.049

=0.01715

4. Poisson distribution formula: P(X=x)= e^-λ. λ^x / x!

e= 2.7183

λ= 50

a.) 2.7183^-50 × 50^0 ÷ 0!

= 5.185 =5

b.) 2.7183^-50 × 50^7 ÷ 7! =

= 5.185 × 7.8125 ÷ 5040

= 8.0368

OPARA PRINCESS ADANNA

2019/245454

1a.) The Poisson distribution: The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time or space.

1b.) Uniform Distribution: A Uniform distribution is a continuous probability distribution and is related to the events which are equally likely to occur.

1c.) The exponential distribution: This is a continuous probability distribution that often concerns the amount of time until some specific event happens

2) N=26

r=6

N-r= 20

n=7

x=4

n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.)

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

1a Poisson probability distributions:

Poisson distribution is a probability that is used to show how many times an event occurs over a specified period. The Poisson distribution is a discreet function, meaning that the variable can only take one specific value in a list. A poisson random variable arise frequently in counting number of occuring events in a given interval of time or space.

b. Uniform probability:

Uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A discrete random variable X, is said to have a uniform distribution if the probability density function of X is defined by P(X=x)= 1/n; x= 1,2…n .

c. Exponential probability distribution:This is a probability distribution of the time between events in a point process that is a process in which events occur continuously and independently at a constant average rate.

2)

r N-r

x n-x

N

n

N=26 r=6 N-r=20 n=7

x=4

6 20

4 3

26

7

6C4 x 20C3 / 26C7

= 17100/657800

= 0.0259

6 20

3 4

26

7

=6C3 x 20C4 / 26C7

= 96900/657800

=0.14731

3) p= 0.35

q= 0.65

x= 7

pqx= 0.35 (0.65)7

= 0.35 x 0.0490

=0.01715

4)

e-ƛ × ƛx /

x!

P(x=0)

= 0.018

P(x=7)

= 0.018×16384 / 5040

=248.912/5040

=0.059

ONYEMA DIVINE OLUCHI

2019/244390

1a) Poisson distribution is a probability distribution, often used to understand independent events that occur at a constant rate within a given interval of time.

1b) Uniform distribution in statistics is a probability distribution where all outcomes are equally likely.

1c) Exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens

2) N=26

r=6

N-r= 20

n=7

x=4

n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.)

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

=0.0183156388

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

ILAMI BENISON IBOH

2019/241788

1A] POISSON PROBABILITY DISTRIBUTION: Poisson distribution is a probability distribution used in Statistics to show the number of times an event is likely to occur over a specific period. Poisson distribution is a Discrete Function.

1B] UNIFORM PROBABILITY DISTRIBUTION: Uniform Distribution is a probability distribution used in Statistics when every outcome has an equal chance of occurrence. Some Uniform Distribution are Continues (infinite number of outcomes) while others are Discrete (finite number of outcomes)

1C] EXPONENTIAL PROBABILITY DISTRIBUTION: Exponential distribution is a probability distribution in which events occur independently and continuously at a constant average rate.

2] N=26

r=6

N-r= 20

n=7

x=4

n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.]

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.]

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

SAMUEL FRANCESS KENILE

2019/250034

1a.) The Poisson distribution is a discrete probability function that means the variable can only take specific values in a given list of numbers, probably infinite.

1b.) The uniform distribution defines equal probability over a given range for a continuous distribution.

1c.) In probability theory, the exponential distribution is defined as the probability distribution of time between events in the Poisson point process

2) N=26

r=6

N-r= 20

n=7

x=4

n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.)

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.)

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

MOETEKE EBELE LOUISA

2019/244608

1a.] Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution.

1b.] Uniform distribution is defined as the type of probability distribution where all outcomes have equal chances or are equally likely to happen and can be bifurcated into a continuous and discrete probability distribution.

1c.] The exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens.

2.] N=26

r=6

N-r= 20

n=7

x=4

n-x=3

P(x)= (rCx)(N-rCn-x)/NCn

(6C4)(20C3)/26C7=(15(1140))/657800

=17100/657800

=0.02599574339

≈0.026

3.]

P(x)=pq^x

Where x=n-1 and q= 1-p

Therefore p(x)= p(1-p)^(n-1)

P = 0.35

n =8 Therefore x= 8-1 =7

q=1-p=1-0.35=0.65

= 0.35(0.65)^7

=0.01715779762

≈0.0172

4.]

P(k)=( λ^k )(e^(-λ) )/k!

λ=50/12=4.1667≈4

When x=0

((4^0 ) e^(-4))/0!

=e^(-4).

≈0.01832

When x=7

(4^7 )(e^(-4) )/7!

=16384(0.01832)/5040

=300.0834276/5040

=0.05954036261

≈0.05954

Name : Nnaji Kelechi

Reg No: 2019/245744

Email: nnajikelechi2001@gmail.com

ANSWERS

NO. 1

POISSON PROBABILITY DISTRIBUTION:

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

CONTINUOUS UNIFORM PROBABILITY DISTRIBUTION:

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.[1] The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)). Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution’s support.

EXPONENTIAL PROBABILITY DISTRIBUTION:

The exponential distribution is a continuous distribution that is commonly used to measure the expected time for an event to occur. For example, in physics it is often used to measure radioactive decay, in engineering it is used to measure the time associated with receiving a defective part on an assembly line, and in finance it is often used to measure the likelihood of the next default for a portfolio of financial assets. It can also be used to measure the likelihood of incurring a specified number of defaults within a specified time period.

NO2

P(X=4doctors and 3nurses)

N=6+20=26

N=7

R=6

P(X=x)=(r¦x)((N-r)¦(n-x))/((N¦n) )

P(X=4doctors and 3nurses))=(6¦4)((26-6)¦(7-4))/((26¦7) )= P(X=x)=(6¦4)(20¦3)/((26¦7) )

=(6∁4*20∁3)/(26∁7)= 0.0260

NO3

P(X=x) = 〖Pq〗^x

P=0.35

q=1-0.35=0.65

P(X=x)=0.35*〖(0.65)〗^7

=0.01716

NO. 4A

P(X=x)= (〖 λ〗^x е^(-λ))/x!

e =2.1782

λ=50/12=4.16666~4

λ=4

P(X=0)=(4^0 (〖2.178) 〗^(-4))/0!=0.04444

NO. 4B

When x =7

P(X=7)=(4^7 (〖2.178) 〗^(-4))/7!=(16384*0.04444)/5040

=0.1445

Daniel Unique AgbenuThe notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

2019/246710

uniquedaniel08@gmail.com

1.Write short notes on the following types of probability distributions

a. Poission probability distribution

Poisson distribution is used when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume or such area more than one occurrence of an event can occur. is used to show how many times an event is likely to occur over a specified period. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. Poisson distribution can be used to estimate how likely it is that something will happen “X” number of times.

Poisson distribution Formula

P(X=x) = e-λ. λx/x!

Where e is Euler’s number (e = 2.71828…)

x is the number of occurrences or number of events in an area of opportunity.

x! is the factorial of x

λ is equal to the expected value or expected number of events(EV) of x when that is also equal to its variance

b. Uniform Probability Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. Under a uniform distribution, each value in the set of possible values has the same possibility of happening.A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.Uniform distribution is a probability distribution that asserts that the outcomes for a discrete set of data have the same probability.

In a discrete uniform distribution, outcomes are discrete and have the same probability.In a continuous uniform distribution, outcomes are continuous and infinite.In a normal distribution, data around the mean occur more frequently.

The probability density function is f(x)=

1/b−a for a ≤ x ≤ b.

c. Exponential Probability Distribution.

The exponential distributidon is often concerned with the amount of time until some specific event occurs.Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

2. Using the hypergeometric method to solve the equation

Formula isP(X=x) = [rCx] [N-rn-x] ÷[NCn]

Where x= number of successes in the number of trials n

N= total number of elements in the population

r= number of successes in the population

N-r = numbe of failures in the population

n= number of trials(sample size)

N= 6+20=26 r=6 n=7 x=4 N-r =20. n-x=3

6C4*20C3÷26C7. = 15*1,140÷675,800

= 17,100÷675,800

=0.025303

3. Using the geometric distribution method

P(X=x) = P(1-P) raise to power x or pqraise to power x

P =0.35. ( 1-P) =1-0.35=0.65

x will be the number of her relationships before marriage

P(X=7) = (0.35*(0.65)*7

= 0.35*0.049022 = 0.01757

4. Using the Poisson distribution method

P(X=x) = e-λ. λx/x!

Where e is Euler’s number (e = 2.71828…)x is the number of occurrences or number of events in an area of opportunity.

x! is the factorial of x

λ is equal to the expected value or expected number of events(EV) of x when that is also equal to its variance

λ to the nearest whole number = 50÷12= 4.166= 4

For no cholocate: x=0

e raise to the power of -4*4raise to power 0÷0!

= 0.0183*1÷1= 0.0183

For 7 chocolate chips;x=7

λ= 4 = 0.0183 *4raise to power 7÷ 7!

=0.0183*16,384÷5,040

=299.8272÷5040

= 0.05948

Name: Ezeh Keren Kamarachi

Reg No: 2019/244045

Department: Economics

Email: kerenezeh@gmail.com

1i) Poisson Probability Distribution

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. … The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

The Poisson distribution, named after French mathematician Siméon Denis Poisson, can be used to estimate how many times an event is likely to occur within “X” periods of time.

Poisson distributions are used when the variable of interest is a discrete count variable.

Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

The Formula for the Poisson Distribution Is

P(X=x)=λ^x.e^-λ /x!

Where:

e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of x when that is also equal to its variance

1ii) Uniform Probability Distribution

statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

Uniform distributions are probability distributions with equally likely outcomes.In a discrete uniform distribution, outcomes are discrete and have the same probability.In a continuous uniform distribution, outcomes are continuous and infinite.

In a normal distribution, data around the mean occur more frequently.

The frequency of occurrence decreases the farther you are from the mean in a normal distribution.

Understanding Uniform Distribution

There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. There are only 6 possible values to return and nothing in between

Types of Uniform Probability Distribution

1) Discrete distribution

2) Continuous distribution

1 Discrete uniform distribution

In statistics and probability theory, a discrete uniform distribution is a statistical distribution where the probability of outcomes is equally likely and with finite values. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. The possible values would be 1, 2, 3, 4, 5, or 6. In this case, each of the six numbers has an equal chance of appearing. Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6.

number of values is finite. It is impossible to get a value of 1.3, 4.2, or 5.7 when rolling a fair die. However, if another die is added and they are both thrown, the distribution that results is no longer uniform because the probability of the sums is not equal. Another simple example is the probability distribution of a coin being flipped. The possible outcomes in such a scenario can only be two. Therefore, the finite value is 2.

2 Continuous uniform distribution

Not all uniform distributions are discrete; some are continuous. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values. Unlike discrete random variables, a continuous random variable can take any real value within a specified range.

A continuous uniform distribution usually comes in a rectangular shape. A good example of a continuous uniform distribution is an idealized random number generator. With continuous uniform distribution, just like discrete uniform distribution, every variable has an equal chance of happening. However, there is an infinite number of points that can exist

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x. The probability density function is f(x)=1b−a f ( x ) = 1 b − a for a ≤ x ≤ b.

1iii) Exponential Probability Distribution

In Probability theory and statistics, the exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens. It is a process in which events happen continuously and independently at a constant average rate. The exponential distribution has the key property of being memoryless. The exponential random variable can be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the supermarket follows an exponential distribution.

The exponential distribution is a continuous probability distribution used to model the time elapsed before a given event occurs.Sometimes it is also called negative exponential distribution.

The exponential distribution is often used to answer in probabilistic terms questions such as:How much time will elapse before an earthquake occurs in a given region?How long do we need to wait until a customer enters our shop? How long will it take before a call center receives the next phone call? How long will a piece of machinery work without breaking down?

All these questions concern the time we need to wait before a given event occurs. If this waiting time is unknown, it is often appropriate to think of it as a random variable having an exponential distribution

2. P(X=4 doctors and 3 nurses), N=26, n=7, r=6

Using hyper geometric formula,

P(X=4 doctors 3 nurses)= [6C4 × (26-6)C(7-4)] ÷ (26C7)

=[15×1140]÷ (657800)

= 0.025995743

3. P(X=x)= p(1-p)^x or pq^x

Using geometric approach,

Where, p= 0.35 and x= 8-1=7

Let x be the number of relationship before the marriage

P(X=7)= 0.35×(1-0.35)^7

=0.35×(0.65)^7

=0.35×0.049022278

=0.017157797

4. f(x) = P(X=x) = (e^-λ* λ^x )/x!

Using poisson formula,

Where, e=2.7183, λ= 50÷12= 4.167and x= number of occurrence

a. P(X=0)= [(4.167^0)× 2.7183^(-4.167)] ÷ 0!

= 0.015498254

b. P(X=7)= [(4.167^7)×(2.7183^-4.167)]÷ 7!

= [(21815.46644)×(0.015498254)] ÷ 5040

=338.1016571÷5040

=0.067

EGWUONWU OLISAEMEKA ELOCHUKWU

2019/245027

ECONOMICS

1a) POISSON DISTRIBUTION; This is a type of probability distribution that is used to show many times an event is likely to occur over a specified period. This is also a type of a discrete count distribution. This Poisson distribution is mostly used to understand independent events that occur at a constant rate within a given interval of time. It was named after a French mathematician Simeon Denis Poisson. For Poisson distribution, the variable can only take integers (ie o, 1, 2, 3 etc.) which means it can’t take fraction or decimal. The application of this distribution can help managers to introduce optimal scheduling system that would not with, say, a normal distribution. It is also used to model financial count data, and many others.

Formula for Poisson distribution; (e –λ λx)/x!

b) UNIFORM DISTRIBUTION; The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When solving problem that has uniform distribution, it is important to note whether the variable is inclusive or exclusive.

Uniform distribution could be discrete or continuous and variables always have the same probability. (eg. A toss of a coin and a roll of a dice)

Formula for a discrete uniform distribution;

Px=1/n

c) EXPONENTIAL DISTRIBUTION; This is a type of continuous distribution that is commonly used to measure the expected time for an event to occur. It is often used to model the time elapsed between events. One of a property of exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution.

Exponential distribution is commonly present when dealing with events that are quickly changing early on.

2) P(X=x); ( rCx N-rCn-r)/(NCn)

P(x=4 doctors & 3 nurses)

N=26, n=7, r=6 ,x=4

ie (6C4 20C1)/(26C7)

=(15*20)/(657800)

=300/657800

=0.000456

3) P(X=x); pqx

X=8-1=7, p=0.35, q=1-0.35=0.65

=0.35(0.65)7

=0.35(0.04902)

=0.01716

4) P(X=x); (e –λ λx)/x!

λ =50/12=4.16

4a)x=o; (e-4 40)/4!

=(0.01832*1)/24

=0.01832/24

=0.000763

4b)x=7; (e-4 47)/7!

=(0.01832*16384)/5040

=300.15/5040

=0.0596

Ogbodo peace chinenyenwa

2017/249543

Nenyepeace2010@gmail.com

Peacenenye.blogspot.com

1.

A: POISSON DISTRIBUTION

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. Poisson distribution is also applied when we want to ascertain the number of times an event occurs in a given area if opportunity.

The formular for Poisson distribution can be written as:

f(x;μ)= (μ^x°e^−μ)/x!

where x=0, 1, … represents the discrete random variable, such as ADC counts recorded by a detection system, and μ>0 is the mean.

B: THE UNIFORM DISTRIBUTIONThe uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

The probability density function is

f(x)=1/b−a

for a ≤ x ≤ b.

For this example, X ~ U(0, 23) and

f(x)=1/23−0

for 0 ≤ X ≤ 23.

Formulas for the theoretical mean and standard deviation are

μ = a+b/2 and σ = √(b−a)^2/12

C: THE EXPONENTIAL DISTRIBUTION

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25).

The probability density function is f(x) = me–mx. The number e = 2.71828182846… It is a number that is used often in mathematics. Scientific calculators have the key “ex.” If you enter one for x, the calculator will display the value

2. Solution:

P(X = 4doctors and 3nurses)

N = 6+20=26; n =7; r =6

(rCx * N-RCn-x)/NCn

P = (6C6*20C1)/26C7

P= 20/657800= 0.0000304

3.solution:

P(X=7) = 0.35*(1-0.35)^7

= 0.0172

4. Solution:

P(X=0) = e^-50 *(50)^0/0!

=1.928[×10^-22]…………1

P(X=7)= e^-50 * 50^7/7!

=2.988[×10^-14]…………..2

From the results above 1 and 2 shows that the answer are in binary forms.

1a. Poisson Distribution: This is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In other words, it is used when you are interested in the number of times an event occurs in a given area of opportunity(i.e a continous unit, internaval of time, volume or any area where an event occurs more than once).

In finance, the poisson distribution could be used to model the expected arrival of orders at specified trading venues. In this case, the poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates.

1b. Uniform Distribution: This is a type of continous probability distribution. It refers to a type of probability distribution in which all outcomes are equally likely. For example, if you randomly approach a person and try to guess his/her birthday, the probability of his/her birthday falling exactly on the date you have guessed follows a uniform distribution. This is because every day of the year has equal chances of being his/her birthday or every day of the year is equally likely to be his/her birthday.

Uniform probability distribution is also called a rectangular distribution. Its distribution is defined by two parameters; a and b. Where a is the minimum and b is the maximum, written as: U(a,b). However, there are two types of uniform probability distribution; continous and discrete uniform probability distributions.

1c. Exponential Distribution: This is a type of continous probability distribution. It is defind as the probability distribution of time between events in the poisson point process. It is concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts.

The exponential distribution can be said to be the contious analogue of the geometric distribution. It is one of the widely used continous distribution.

2. Using the hypergeometric distribution;

P(X=x) = ( r combination x × N – r combination n – x )/ N combination n .

Where;

N = 26

n = 7

r = 6

Therefore;

i. P(X = 4) = (6 com. 4 × 26 – 6 com. 7- 4) / 26 com. 7

= (6 com. 4 × 20 com. 3 )/ 26 com. 7

= (15 × 1140)/657800

= 17100/657800

= 0.026

Hence, P(X = 4) = 0.026

ii. P(X = 3) = (6 com. 3 × 20 com. 4)/ 26 com. 7

= (20 × 4845)/657800

= 96900/657800

= 0.147

Hence, P(X = 3) = 0.147

3. Using the geometric distribution;

P(X = x) = P(1 – P)^x

Where;

P = 0.35

1 – P = 0.65

Therefore;

P(X = 7) = 0.35(0.65)^7

= 0.35(0.0490)

= 0.01715

Hence, P(X = 7) = 0.01715

4. Using poisson distribution;

P(X=x)=(λ^x × e^-λ)/x!

Where;

λ = 12

Therefore;

i. P(X = 0) = 12^0 × e^-12/0!

= 0.000006144

ii. P(X = 7) = (12^7 × e^-12)/7!

= 0.04368

Hence, P(X = 7) = 0.04368

EGWUONWU OLISAEMEKA ELOCHUKWU

2019/245027

ECONOMICS

1a) POISSON DISTRIBUTION; This is a type of probability distribution that is used to show many times an event is likely to occur over a specified period. This is also a type of a discrete count distribution. This Poisson distribution is mostly used to understand independent events that occur at a constant rate within a given interval of time. It was named after a French mathematician. For Poisson distribution, the variable can only take integers (ie o, 1, 2, 3 etc.) which means it can’t take fraction or decimal. The application of this distribution can help managers to introduce optimal scheduling system that would not with, say, a normal distribution. It is also used to model financial count data, and many others.

Formula for Poisson distribution; (e –λ λx)/x!

b) UNIFORM DISTRIBUTION; The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When solving problem that has uniform distribution, it is important to note whether the variable is inclusive or exclusive.

Uniform distribution could be discrete or continuous and variables always have the same probability. (eg. A toss of a coin and a roll of a dice)

Formula for a discrete uniform distribution;

Px=1/n

c) EXPONENTIAL DISTRIBUTION; This is a type of continuous distribution that is commonly used to measure the expected time for an event to occur. It is often used to model the time elapsed between events. One of a property of exponential distribution is that it can be viewed as a continuous analogue of the geometric distribution.

Exponential distribution is commonly present when dealing with events that are quickly changing early on.

2) P(X=x); ( rCx N-rCn-r)/(NCn)

P(x=4 doctors & 3 nurses)

N=26, n=7, r=6 ,x=4

ie (6C4 20C1)/(26C7)

=(15*20)/(657800)

=300/657800

=0.000456

3) P(X=x); pqx

X=8-1=7, p=0.35, q=1-0.35=0.65

=0.35(0.65)7

=0.35(0.04902)

=0.01716

4) P(X=x); (e –λ λx)/x!

λ =50/12=4.16

4a)x=o; (e-4 40)/4!

=(0.01832*1)/24

=0.01832/24

=0.000763

4b)x=7; (e-4 47)/7!

=(0.01832*16384)/5040

=300.15/5040

=0.0596

Arinze ebuka KelvinIn statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

2019/246530

Economics major

Uniform Distribution Definition

Uniform distribution is a type of probability distribution in which all outcomes are equally likely.

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution.

P(×=7)=4.17^7×£-^4.17/7!=21=25.6453×0.01545226/5,040 =338.8007752/504=0.067.

P=0.35,q=1-0.35 x=7

Pqx.(×=7)=0.35×0.65^7

= 0.35×0.0490222787=0.0172

P(x=0)=4.17.2.7183/0!=

1×0.01545226

P(x=0)=0.0155

NAME; EZUGWU JOHNSON CHINECHEREM

REG NO; 2019/245390

NUMBER ONE:

POISSON PROBABILITY DISTRIBUTION

A Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals. The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space). If we let X = the number of events in a given interval. Then, if the mean number of events per interval is λ the probability of observing x events in a given interval is given by

P(X = x) = e−λ λ/ x!

• e is Euler’s number (e = 2.71828…)

• x is the number of occurrences

• x! is the factorial of x

• λ is equal to the expected value (EV) of x when that is also equal to its variance.

UNIFORM PROBABILITY DISTRIBUTION

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b) . Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution’s support.

EXPONENTIAL PROBABILITY DISTRIBUTION

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less. In addition to being used for the analysis of Poisson point processes. The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others

NUMBER TWO

P(x=4doctors and 3 nurses)

N= 6 + 20 =26

N = 7, r = 6

P(X =X) (rcx *N-r(n-x)/NCn

P(x=4 doctors and 3 nurses)

=( 6 c4 * 20c3)/26c7

=(15 *1140)/657800

=0.02599574339

NUMBER THREE

P(X=X) = PQx

P =0.35 Q = 1-P =0.65

Before she got married at 8th relationship

P(x =7) 0.35 *(0.65)7= 0.01715779762

NUMBER FOUR

(A) No chocolate chips

P(X = x) =( e−λ λx)/ x!

e = 2.71828…

x =0

λ = 50/12 =4.166666667 = 4

P(X = 0) (40*(2.7183)-4/0! =

[1*(2.7183)-4]/1 = 0.001831514914

(B) 7 chocolate

e = 2.71828…

x =7

λ = 50/12 =4.166666667 = 4

P(X = 7) (47*(2.7183)-4/7! =

(16384*(2.7183)-4]/7! =

[16384*(2.7183)-4]/5040 = 0.05953877054

NAME; EZUGWU JOHNSON CHINECHEREM

REG NO; 2019/245390

NUMBER ONE:

POISSON PROBABILITY DISTRIBUTION

A Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals. The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space). If we let X = the number of events in a given interval. Then, if the mean number of events per interval is λ the probability of observing x events in a given interval is given by

P(X = x) = e−λ λ/ x!

• e is Euler’s number (e = 2.71828…)

• x is the number of occurrences

• x! is the factorial of x

• λ is equal to the expected value (EV) of x when that is also equal to its variance.

UNIFORM PROBABILITY DISTRIBUTION

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b) . Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution’s support.

EXPONENTIAL PROBABILITY DISTRIBUTION

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less. In addition to being used for the analysis of Poisson point processes. The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others

NUMBER TWO

P(x=4doctors and 3 nurses)

N= 6 + 20 =26

N = 7, r = 6

P(X =X) (rcx *N-r(n-x)/NCn

P(x=4 doctors and 3 nurses)

=( 6 c4 * 20c3)/26c7

=(15 *1140)/657800

=0.02599574339

NUMBER THREE

P(X=X) = PQx

P =0.35 Q = 1-P =0.65

Before she got married at 8th relationship

P(x =7) 0.35 *(0.65)7= 0.01715779762

NUMBER FOUR

(A) No chocolate chips

P(X = x) =( e−λ λx)/ x!

e = 2.71828…

x =0

λ = 50/12 =4.166666667 = 4

P(X = 0) (40*(2.7183)-4/0! =

[1*(2.7183)-4]/1 = 0.001831514914

(B) 7 chocolate

e = 2.71828…

x =7

λ = 50/12 =4.166666667 = 4

P(X = 7) (47*(2.7183)-4/7! =

(16384*(2.7183)-4]/7! =

[16384*(2.7183)-4]/5040 = 0.05953877054

Name: Onyia Ugochukwu Sullivan

Reg No: 2019/249490

Department: Economics

Email Address: Ugoski135@gmail.com

ASSIGNMENT:

1)

POISSON PROBABILITY DISTRIBUTION is a discrete probability distribution that is used when you are interested on the number of times an event occurs in a given area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here, there can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes.

FORMULA: (PX=x) = e^-*ƛ ^x/x!

UNIFORM PROBABILITY DISTRIBUTION is a continuous probability distribution that is distributed on interval. It has special use in rounding errors when measurement are recorded to some degree of accuracy. The application of this distribution is based on the assumption that the probability of falling in an interval of fixed length within (α, ß) is constant.

EXPONENTIAL PROBABILITY DISTRIBUTION is a random variable x is said to have an exponential distribution with parameter ß if the probability density of x is defined by f(x) = (e^x/ ß )/ ß x>0. Exponential distribution is used in practical problem to represent the distribution of time that elapse before the occurrence of some event, for instance, the time required to serve a customer at some service facility

2i) Using Hypergeometric distribution:

Where

r = 6

n = 7

N = 26

X = 4

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ4]*[26-6ʗ7-4]/[26ʗ7]

[6ʗ4]*[20ʗ3]/[26ʗ7]

=0.02599574339

ii)

Where

r = 6

n = 7

N = 26

X = 3

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ3]*[26-6ʗ7-3]/[26ʗ7]

[6ʗ3]*[20ʗ4]/[26ʗ7]

=0.1473092125

3) Using geometric distribution => pqX

Where

P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

ii)

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

NAME: ASOGWA REJOICE CHINECHEREM

Reg no; 2019/242727

1. Write short notes on:

*Poisson Probability Distribution*

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates.

*Uniform Probability Distribution*

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints.

*Exponential Probability Distribution*

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length of time, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts.

Solution3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number. What is the probability that it contains;

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp (final answer)

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.(final answer)

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595(final answer)

Name: Ogbuagu Chiamaka Rosita

Reg No: 2019/241915

Department: Economics

Email Address: chiamakaogbuagu.05@gmail.com

ANSWERS TO THE ASSIGNMENT: QUESTION 1

POISSON PROBABILITY DISTRIBUTION: is a type of discrete probability distribution that is used when you are interested on the number of times an event occurs in a given area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here, there can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes.

FORMULA: (PX=x) = e^-ᨂ* ᨂ^x/x!

UNIFORM PROBABILITY DISTRIBUTION: is a type of continuous probability distribution that is distributed on interval. It has special use in rounding errors when measurement are recorded to some degree of accuracy. The application of this distribution is based on the assumption that the probability of falling in an interval of fixed length within (α, ß) is constant.

EXPONENTIAL PROBABILITY DISTRIBUTION: a random variable x is said to have an exponential distribution with parameter ß if the probability density of x is defined by f(x)= (e^x/ ß )/ ß x>0. Exponential distribution is often used in practical problem to represent the distribution of time that elapse before the occurrence of some event, for instance, the time required to serve a customer at some service facility.

2i) Using Hypergeometric distribution:

Where

r = 6

n = 7

N = 26

X = 4

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ4]*[26-6ʗ7-4]/[26ʗ7]

[6ʗ4]*[20ʗ3]/[26ʗ7]

=0.02599574339

ii)

Where

r = 6

n = 7

N = 26

X = 3

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ3]*[26-6ʗ7-3]/[26ʗ7]

[6ʗ3]*[20ʗ4]/[26ʗ7]

=0.1473092125

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490222789062

= 0.0171577976171

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

Name: Aniukwu Chisom Sylvia

Reg No: 2019/243386

Department: Economics

Email Address: aniukwuchisomsylvia2002@gmail.com

ANSWERS TO THE ASSIGNMENT: QUESTION 1

POISSON PROBABILITY DISTRIBUTION. This is a type of discrete probability distribution that is used when you are interested on the number of times an event occurs in a given area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here, there can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes.

FORMULA: (PX=x) = e^-ᨂ* ᨂ^x/x!

UNIFORM PROBABILITY DISTRIBUTION. This is a type of continuous probability distribution that is distributed on interval. It has special use in rounding errors when measurement are recorded to some degree of accuracy. The application of this distribution is based on the assumption that the probability of falling in an interval of fixed length within (α, ß) is constant.

EXPONENTIAL PROBABILITY DISTRIBUTION. This is another type of continuous distribution, a random variable x is said to have an exponential distribution with parameter ß if the probability density of x is defined by f(x)= (e^x/ ß )/ ß x>0. Exponential distribution is used in practical problem to represent the distribution of time that elapse before the occurrence of some event, for instance, the time required to serve a customer at some service facility

2i) Using Hypergeometric distribution:

Where

r = 6

n = 7

N = 26

X = 4

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ4]*[26-6ʗ7-4]/[26ʗ7]

[6ʗ4]*[20ʗ3]/[26ʗ7]

=0.02599574339

ii)

Where

r = 6

n = 7

N = 26

X = 3

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ3]*[26-6ʗ7-3]/[26ʗ7]

[6ʗ3]*[20ʗ4]/[26ʗ7]

=0.1473092125

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

Name: Ogbaga Stella Chinwendu

Reg No: 2019/241733

Department: Economics

Email Address: ogbagachinwendu@gmail.com

ANSWERS TO THE ASSIGNMENT: QUESTION 1

POISSON PROBABILITY DISTRIBUTION: This is used when you are interested on the number of times an event occurs in a given area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. There can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes.

FORMULA: (PX=x) = e^-ᨂ* ᨂ^x/x!

UNIFORM PROBABILITY DISTRIBUTION: is a type of continuous probability distribution that is distributed on interval. It has special use in rounding errors when measurement are recorded to some degree of accuracy. The application of this distribution is based on the assumption that the probability of falling in an interval of fixed length within (α, ß) is constant.

EXPONENTIAL PROBABILITY DISTRIBUTION: a random variable x is said to have an exponential distribution with parameter ß if the probability density of x is defined by f(x)= (e^x/ ß )/ ß x>0. Exponential distribution is used in practical problem to represent the distribution of time that elapse before the occurrence of some event, for instance, the time required to serve a customer at some service facility

2i) Using Hypergeometric distribution:

Where

r = 6

n = 7

N = 26

X = 4

[6ʗ4]*[26-6ʗ7-4]/[26ʗ7]

[6ʗ4]*[20ʗ3]/[26ʗ7]

=0.02599574339

ii)

Where

r = 6

n = 7

N = 26

X = 3

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ3]*[26-6ʗ7-3]/[26ʗ7]

[6ʗ3]*[20ʗ4]/[26ʗ7]

=0.1473092125

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

4) ƛx e-x

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

Name: Anolue Chinecherem Stephanie

Reg No: 2019/244424

Department: Economics

Email Address: anoluesteph2002@gmail.com

ANSWERS TO THE ASSIGNMENT: QUESTION 1

POISSON PROBABILITY DISTRIBUTION: This refers to a type of discrete probability distribution that is used when you are interested on the number of times an event occurs in a given area of opportunity is a continuous unit or interval of time such area in which more than one occurrence of an event can occur. Here, there can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes.

Formula for the poisson probability distribution: (PX=x) = e^-ᨂ* ᨂ^x/x!

UNIFORM PROBABILITY DISTRIBUTION is distributed on interval. It has special use in rounding errors when measurement are recorded to some degree of accuracy. The application of this distribution is based on the assumption that the probability of falling in an interval of fixed length within (α, ß) is constant.

EXPONENTIAL PROBABILITY DISTRIBUTION: a random variable x is said to have an exponential distribution with parameter ß if the probability density of x is defined by f(x)= (e^x/ ß )/ ß x>0. Exponential distribution is used in practical problem to represent the distribution of time that exhausts before the occurrence of some event, for instance, the time required to serve a customer at some service facility

2i) Using Hypergeometric distribution:

Where

r = 6

n = 7

N = 26

X = 4

[6ʗ4]*[26-6ʗ7-4]/[26ʗ7]

[6ʗ4]*[20ʗ3]/[26ʗ7]

=0.02599574339

ii)

Where

r = 6

n = 7

N = 26

X = 3

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ3]*[26-6ʗ7-3]/[26ʗ7]

[6ʗ3]*[20ʗ4]/[26ʗ7]

=0.1473092125

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490

= 0.0172

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

NAME: ADIGWE ANTHONY CHIBUIKEM

REG. NO: 2019/245463

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The formula is:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

c) Exponential probability distribution:The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time from now until an earthquake occurs has an exponential distribution. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The formula is stated below:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqxBefore she got married at the 8th relationship, she failed in the first seven relationships. So x is 7. Therefore,

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

NAME; IGBADI ODIYA DANLADI

REG NO; 2019/244347

NUMBER ONE:

POISSON PROBABILITY DISTRIBUTION

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. … The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. It is denoted as

P(X = x) = e−λ λ/ x!

• e is Euler’s number (e = 2.71828…)

• x is the number of occurrences

• x! is the factorial of x

• λ is equal to the expected value (EV) of x when that is also equal to its variance.

UNIFORM PROBABILITY DISTRIBUTION

In statistics, in uniform distribution refers to a type of probability distribution which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely.

EXPONENTIAL PROBABILITY DISTRIBUTION

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution.

NUMBER TWO

P(x=4doctors and 3 nurses)

N= 6 + 20 =26

N = 7, r = 6

P(X =X) (rcx *N-r(n-x)/NCn

P(x=4 doctors and 3 nurses)

=( 6 c4 * 20c3)/26c7

=(15 *1140)/657800

=0.02599574339

NUMBER THREE

P(X=X) = PQx

P =0.35 Q = 1-P =0.65

Before she got married at 8th relationship

P(x =7) 0.35 *(0.65)7= 0.01715779762

NUMBER FOUR

(A) No chocolate chips

P(X = x) =( e−λ λx)/ x!

e = 2.71828…

x =0

λ = 50/12 =4.166666667 = 4

P(X = 0) (40*(2.7183)-4/0! =

[1*(2.7183)-4]/1 = 0.001831514914

(B) 7 chocolate

e = 2.71828…

x =7

λ = 50/12 =4.166666667 = 4

P(X = 7) (47*(2.7183)-4/7! =

(16384*(2.7183)-4]/7! =

[16384*(2.7183)-4]/5040 = 0.05953877054

NAME: Oguaju Adaobi Dorothy

REG. NO: 2016/235292

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution: The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates. Poisson distributions are very useful for smart order routers and algorithmic trading. The formula is stated below:

{ P(X=x) = (λxe-λ) / x! }

where, e is the base of the natural logarithm system (2.71828..)

x is the number of events in an area of opportunity

x! is the factorial of x

λ is the expected number of events

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution: The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive. The formula is: { Px = 1 / n }

where, Px = probability of a discrete value n = number of values in the range

c) Exponential probability distribution: The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time from now until an earthquake occurs has an exponential distribution. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The formula is stated below:

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ^2)^0.5

3) P(X=x) = pq^x p=0.35, q=1-p=0.65

Before she got married at the 8th relationship, she failed in the first seven relationships. So x is 7.

Therefore, P(x=7)=0.35 *(0.65)7 = 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0! =

[ 1 * (2.7183)-4 ] / 1 = 0.01831514914

b) 7 chocolate chips. P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7! = [ 16384 * (2.7183)-4 ] / 5040 = 0.05953877054

Name : NKWOCHA IKECHUKWU BONAVENTURE

Reg No: 2017/249530

Email: Ikechukwubonaventure63@gmail.com

ANSWERS

NO. 1

Discussion on the following:

POISSON PROBABILITY DISTRIBUTION

CONTINUOUS UNIFORM PROBABILITY DISTRIBUTION

CONTINUOUS UNIFORM PROBABILITY DISTRIBUTION

POISSON PROBABILITY DISTRIBUTION:

It was named after French mathematician Siméon Denis Poisson.

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

P(X=x) = ℮ˉ۶ ٨͋/x! x = 0, 1, 2 …n

CONTINUOUS UNIFORM PROBABILITY DISTRIBUTION:

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.[1] The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)). Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution’s support.

P(X=x) = 1/n x = 1,2, …,n

EXPONENTIAL PROBABILITY DISTRIBUTION:

The exponential distribution is a continuous distribution that is commonly used to measure the expected time for an event to occur. For example, in physics it is often used to measure radioactive decay, in engineering it is used to measure the time associated with receiving a defective part on an assembly line, and in finance it is often used to measure the likelihood of the next default for a portfolio of financial assets. It can also be used to measure the likelihood of incurring a specified number of defaults within a specified time period.

NO2

P(X=4doctors and 3nurses)

N=6+20=26

N=7

R=6

P(X=x)=

P(X=4doctors and 3nurses)== P(X=x)=

== 0.0260

NO3

P(X=x) =

P=0.35

q=1-0.35=0.65

P(X=x)=0.35

=0.01716

NO. 4A

P(X=x)=

e =2.1782

λ=

λ=4

P(X=0)=

NO. 4B

When x =7

P(X=7)=

=0.1445

Name : OSUIWU ADIMCHINOBI PƐACE

Reg No: 2017/249570

Email: osuiwuadimchi@gmail.com

ANSWERS

NO. 1

Discussion on the following:

POISSON PROBABILITY DISTRIBUTION:

Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

CONTINUOUS UNIFORM PROBABILITY DISTRIBUTION:

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.[1] The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (e.g. [a, b]) or open (e.g. (a, b)). Therefore, the distribution is often abbreviated U (a, b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally.

P(X=x) = 1/n x = 1,2, …,n

EXPONENTIAL PROBABILITY DISTRIBUTION:

The exponential distribution is a continuous distribution that is commonly used to measure the expected time for an event to occur. For example, in physics it is often used to measure radioactive decay, in engineering it is used to measure the time associated with receiving a defective part on an assembly line, and in finance it is often used to measure the likelihood of the next default for a portfolio of financial assets. It can also be used to measure the likelihood of incurring a specified number of defaults within a specified time period.

NO2

P(X=4doctors and 3nurses)

N=6+20=26

N=7

R=6

P(X=x)=

P(X=4doctors and 3nurses)== P(X=x)=

== 0.0260

NO3

P(X=x) =

P=0.35

q=1-0.35=0.65

P(X=x)=0.35

=0.01716

NO. 4A

P(X=x)=

e =2.1782

λ=

λ=4

P(X=0)=

NO. 4B

When x =7

P(X=7)=

=0.1445

NAME: Udekwu Sharon Chika

REG. NO: 2019/249132

DEPARTMENT: ECONOMICS

1a) Poisson probability distribution:

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson. The probability density function is given as;

{ P(X=x) = (λxe-λ) / x! }

where, e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of x when that is also equal to its variance

The mean = λ

The variance = λ

The standard deviation = λ0.5

b) Uniform probability distribution:

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same. The probability density function is given as:

{ Px = 1 / n }

where, Px = probability of a discrete value

n = number of values in the range

The mean = 0

The variance = 1

The standard deviation = 1

c) Exponential probability distribution:

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes, it is found in various other contexts. The probability density function is given as;

{ f(x)= λe- λx }

The mean = 1/ λ

The variance =1/ λ2

The standard deviation =(1/ λ2)0.5

2) P(x=4 doctors and 3 nurses)

N=6+20=26

n=7, r=6

P(X=x)= ( rCx * N-rCn-x ) / NCn

P(x=4 doctors and 3 nurses) = ( 6C4 * 20C3 ) / 26C7

=( 15 *1140 ) / 657800

= 0.02599574339

3) P(X=x) = pqx

p=0.35, q=1-p=0.65

P(x=7)=0.35 *(0.65)7

= 0.01715779762

4a) No chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 0

P(x=0) = [ 40 * (2.7183)-4 ] / 0!

= [ 1 * (2.7183)-4 ] / 1

= 0.01831514914

b) 7 chocolate chips.

P(X=x) = (λxe-λ) / x!

λ = 50 / 12 = 4.166666667 = 4

e = 2.7183, x = 7

P(x=7) = [ 47 * (2.7183)-4 ] / 7!

= [ 16384 * (2.7183)-4 ] / 5040

= 0.05953877054

Achime Chiamaka Nkiru.

2012/181478

Econmics major

Eco 231 assignment

External candidate

Poisson probability distribution shows that poisson random variablewarise frequently in counting number of events in a given interval office or space. Let x be a discrete random variable representing the frequency of occurence of an event in a non overlapping interval of time or space. Then X is said to have a poisson distribution with some parameters if the probability mass function of X is P(X=x).

Uniform probability distribution is a type of distribution in which all the outcomes are equally likely. A deck of card has within it uniform distributions because the likelihood of drawing a heart, a club, a spade are equally likely. It is called a rectangular distribution and there is a constant probability.

Exponential probability distribution has to do with time between 2 events in a poisson point process. A process in which events occur continuously and independently at a constant average rate. It is a particular case of gamma distribution .

Answers

A; Poisson Distribution;

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period.

B; The Uniform DistributionThe uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

The notation for the uniform distribution is X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

The probability density function is f(x)=1b−af(x)=1b−a for a ≤ x ≤ b.

C;. The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.

2; ; A; N=26,n=7 ,r=6, x=4 6C4 X 20C3 /26C7 =17100 /657,800 =0.026

B; 3 nurses P(x=x) =7-3 ;x=4 P(X=4) 6C4x20C3 /26C7 =15 X 1140 /657800 =0.026 0.026X100 =2.6%

3; P=0.35 q=1-p. q=1-0.35=0.65 X=8-1, x=7 P(X=7)= (0.35)(0.65)^7 =0.35 X 0.049 =0.0172 0.0172 X100 =1.7chances.

4;. A; P(X=0) =50^0 X 2.7183^-50 /1 =5.185 =5

B; 50^7 X 2.7183^-50 /7! =7.8125 X 5.185 /5,040 =0.00804

=0.00804 X 100 =0.804 = 1

NAME: MGBOH CHIDERA MARTINS

REG NO: 2019/242146

DEPT: ECONOMICS

COURSE: ECO231

1 a] Poisson probability distribution

A Poisson distribution, named after French mathematician Siméon Denis Poisson, can be used to estimate how many times an event is likely to occur within “X” periods of time. A Poisson distribution is a tool that helps to predict the probability of certain events happening when you know how often the event has occurred. It gives us the probability of a given number of events happening in a fixed interval of time. In Poisson distribution, lambda is the average rate of value for a function.

b] Uniform probability distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. There are two types of uniform distributions: discrete and continuous.

Discrete uniform distribution: In statistics and probability theory, a discrete uniform distribution is a statistical distribution where the probability of outcomes is equally likely and with finite values. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die. The possible values would be 1, 2, 3, 4, 5, or 6. In this case, each of the six numbers has an equal chance of appearing. Therefore, each time the 6-sided die is thrown, each side has a chance of 1/6.

Continuous uniform distribution: Not all uniform distributions are discrete; some are continuous. A continuous uniform distribution (also referred to as rectangular distribution) is a statistical distribution with an infinite number of equally likely measurable values.

The formula for a discrete uniform distribution

Px = 1/n

where:

Px=Probability of a discrete value

n=Number of values in the range

c] Exponential probability distribution

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

2] Six(6) doctors and twenty(20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

Solution

Hypergeometic formula

P(X=x)= [kCx ]•[ N-kCn-x ] / [ NCn ]

Where

N: The number of items in the population

.k: The number of items in the population that are classified as successes.

n: The number of items in the sample.

x: The number of items in the sample that are classified as successes.

Probability that 4 doctors are picked

N=26

n=7

K=6

X=4

P(X=4)= [6C4]•[26–6C7–4] / [26C7]= [6C4]•[20C3] / [26C7] = 15•1140/657800 = 17100/657800=0.0260

Probability that 3 nurse are picked

N=26

N=7

K=20

X=3

P(x=3)=[20C3]•[26–20C7–3] / [26C7]= [20C3]•[6C4] / [26C7]= 1140•15/657800=17100/657800=0.0260

P(4 or 3)= p(4) + p(3) = 0.0260 + 0.0260= 0.052

3] Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

Solution

Geometric formula: f(x) =p• (1 − p)^x − 1 or p•q^x–1

Where

P= probability of success

q= probability of failure which is (1 – p)

P= 0.35

q= 1–0.35= 0.65

x=8

f(x)=0.35•0.65^8–1

f(x)=0.35•0.65^7= 0.35•0.0490= 0.0172

4} To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using Lambda (the mean) to the nearest whole number. What is the probability that it contains:

a] No chocolate chips?

b} 7 chocolate chips?

a] x=0

Lamba=0.50

P(X=0)= 0.50^0•e^—0.50/0!= 1•0.6065/1=0.6 ~ 1

b] X=7

Lamba=0.50

P(X=7) = 0.50^7•e^—0.50/7!=0.0078•0.6065/5040=0.0047/5040=0

NAME:OZONWU CHUKWUEBUKA SILAS

REG NO:2019/244686

POISSION DISTRIBUTION:

Poisson distribution is used in testing how unusual an event frequency is for a given interval.

This is also used when you are interested in the number of times an event occur in a given area of opportunity.An arwa of opportunity is a continuous unit or interval of time here other can be unlimited number of outcome.

WHEN TO APPLY POISSION DISTRIBUTION

When you wish to count number of times an event occur in a given area of opportunity

PROPERTIES OF POSSION DISTRIBUTION

1. The number of event that occur inan area of opportunity is independent of the number of event

2. The probability that two or more event occur in a given area of opportunity approaches zero as the area of opportunity become smaller.

3. The average number of event pee unit denote as λ(Lambda).

POISSON DISTRIBUTION FORMULAR

P(X=x) = (e -λ λ x )/x!

Where

x=number of event in an area of opportunity

λ(Lambda)=expected number of event

e=base of national logarithm.

MEAN μ=λ

VARIANCE σ 2=λ

UNIFORM PROBABILITY DISTRIBUTION

uniform distribution refers to a type of probability distribution in which all outcomes are equally

likely . A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely.

TYPES OF UNIFORM PROBABILITY DISTRIBUTION

1. Discrete uniform probability distribution

2. Continuous uniform probability distribution

discrete probability distribution, The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. There are only 6 possible values to return and nothing in between

Continuous uniform probability distribution, An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there is an infinite number of points between 0.0 and 1.0.

FORMULAR FOR UNIFORM PROBABILITY DISTRIBUTION

P(x) = 1/ (B-A) for A≤ x ≤B.

“A” is the location parameter: The location parameter tells you where the center of the graph is.

“B” is the scale parameter : The scale parameter stretches the graph out on the horizontal axis.

MEAN P(X) = (b + a) / 2.

“a” in the formula is the minimum value in the distribution,

and “b” is the maximum value.

The variance of a uniform random variable is:

VAR(x) = (1/12)(b-a) 2

EXPONENTIAL PROBABILITY DISTRIBUTION

Exponential distribution is usually use when dealing with event that are rapidly changing early on for. Example news blog.

The exponential distribution is a continuous probability distribution used to model the time elapsed before a given event occurs.Just like Poisson they both measure time of occurance of an event but the difference is that in Poisson,it measure the unit of time while exponential measure the time elapsed before a given event occurs.

RELATION BETWEEN POISSON DISTRIBUTION AND EXPONENTIAL

Suppose that An event can occur more than once;

The time elapsed between two successive occurrences is exponentially distributed and independent of previous occurrences.

The number of occurrences of the event within a given unit of time has a Poisson distribution.

ANSWER 2

2. P(x=4 doctor and 3 nurses)

N=26,r=6,n=7.

6C4*20C3/26C7.

=0.025996.

ANSWER 3

3.P(x=7)

P=0.35,q=1-0.35=0.65.

=0.35*0.65^7

0.01716.

ANSWER 4

a. P(x=0)

λ=50/12=4.17

whole number=4

e^-4*4^0/0!

=0.0183.

b. P(x=7)

e^-4*4^7/7!

=0 0595.

NAME:OZONWU CHUKWUEBUKA SILAS

REG NO:2019/244686

POISSION DISTRIBUTION:

Poisson distribution is used in testing how unusual an event frequency is for a given interval.

This is also used when you are interested in the number of times an event occur in a given area of opportunity.An area of opportunity is a continuous unit or interval of time here other can be unlimited number of outcome.

WHEN TO APPLY POISSION DISTRIBUTION

When you wish to count number of times an event occur in a given area of opportunity

PROPERTIES OF POSSION DISTRIBUTION

1. The number of event that occur inan area of opportunity is independent of the number of event

2. The probability that two or more event occur in a given area of opportunity approaches zero as the area of opportunity become smaller.

3. The average number of event pee unit denote as λ(Lambda).

POISSON DISTRIBUTION FORMULAR

P(X=x) = (e -λ λ x )/x!

Where

x=number of event in an area of opportunity

λ(Lambda)=expected number of event

e=base of national logarithm.

MEAN μ=λ

VARIANCE σ 2=λ

UNIFORM PROBABILITY DISTRIBUTION

uniform distribution refers to a type of probability distribution in which all outcomes are equally

likely . A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely.

TYPES OF UNIFORM PROBABILITY DISTRIBUTION

1. Discrete uniform probability distribution

2. Continuous uniform probability distribution

discrete probability distribution, The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. There are only 6 possible values to return and nothing in between

Continuous uniform probability distribution, An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there is an infinite number of points between 0.0 and 1.0.

FORMULAR FOR UNIFORM PROBABILITY DISTRIBUTION

P(x) = 1/ (B-A) for A≤ x ≤B.

“A” is the location parameter: The location parameter tells you where the center of the graph is.

“B” is the scale parameter : The scale parameter stretches the graph out on the horizontal axis.

MEAN P(X) = (b + a) / 2.

“a” in the formula is the minimum value in the distribution,

and “b” is the maximum value.

The variance of a uniform random variable is:

VAR(x) = (1/12)(b-a) 2

EXPONENTIAL PROBABILITY DISTRIBUTION

Exponential distribution is usually use when dealing with event that are rapidly changing early on for. Example news blog.

The exponential distribution is a continuous probability distribution used to model the time elapsed before a given event occurs.Just like Poisson they both measure time of occurance of an event but the difference is that in Poisson,it measure the unit of time while exponential measure the time elapsed before a given event occurs.

RELATION BETWEEN POISSON DISTRIBUTION AND EXPONENTIAL

Suppose that An event can occur more than once;

The time elapsed between two successive occurrences is exponentially distributed and independent of previous occurrences.

The number of occurrences of the event within a given unit of time has a Poisson distribution.

ANSWER 2

2. P(x=4 doctor and 3 nurses)

N=26,r=6,n=7.

6C4*20C3/26C7.

=0.025996.

ANSWER 3

3.P(x=7)

P=0.35,q=1-0.35=0.65.

=0.35*0.65^7

0.01716.

ANSWER 4

a. P(x=0)

λ=50/12=4.17

whole number=4

e^-4*4^0/0!

=0.0183.

b. P(x=7)

e^-4*4^7/7!

=0 0595.

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function meaning that the variable can only take specific values in a (potentially infinite) list Examples include number of mosquito bite on person,number of car accidents over a certain range of a road, number of computer crashes in a day etc.Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution.Uniform probability distribution are probability distribution with equally likely outcome. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

No chocolate

7 chocolates.

P(x=0) ×/0

=0.0067×1/1

=0.0067

P(x=7) ×/5040

= 0.0067/5040

=523.4/5040

=0.1038.

1. Write short notes on the following types of probability distributions:

Poisson probability distribution: it is a discrete function used to show how many times an event is likely to occur over a specified period. Examples include number of mosquito bite on person, number of motor accident in a day etc.

Uniform probability distribution: it is also known as Continuous uniform distribution . It is a family of symmetric probability distributions. This continuous probability distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a continuous probability distribution in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. find the probability that she will get married in her 8th relationship?

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

4. To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda too the nearest whole number. What is the probability that it contains?

No chocolate

7 chocolates.

P(x=0) ×/0

=0.0067×1/1

=0.0067

P(x=7) ×/5040

= 0.0067/5040

=523.4/5040

=0.1038.

UJAM STANLEY CHIBUEZE

2019/242764

1a) Poisson Probability Distribution

In probability theory and statistics, the Poisson distribution French pronunciation: named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.[1] The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

b) Uniform Probability Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) Exponential Probability Distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

Name: Amoke chibuike Anthony

Reg number: 2017/249961

Department: Economics

Email Address: anthonyamoke@gmail.com

NO 1)

POISSON PROBABILITY DISTRIBUTION: is a type of discrete probability distribution that is used when you are interested on the number of times an event occurs in a given area of opportunity is a continuous unit or interval of time, volume or such area in which more than one occurrence of an event can occur. Here, there can be unlimited number of possible outcomes unlike in binomial distribution where we have only two possible outcomes.

FORMULA: (PX=x) = e^-ᨂ* ᨂ^x/x!

UNIFORM PROBABILITY DISTRIBUTION: is a type of continuous probability distribution that is distributed on interval. It has special use in rounding errors when measurement are recorded to some degree of accuracy. The application of this distribution is based on the assumption that the probability of falling in an interval of fixed length within (α, ß) is constant.

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less.

2i) Using Hypergeometric distribution:

Where

r = 6

n = 7

N = 26

X = 4

[6ʗ4]*[26-6ʗ7-4]/[26ʗ7]

[6ʗ4]*[20ʗ3]/[26ʗ7]

=0.02599574339

ii)

Where

r = 6

n = 7

N = 26

X = 3

P(X-x)= [rʗx]*[N-rʗ n-x]/[Nʗn]

[6ʗ3]*[26-6ʗ7-3]/[26ʗ7]

[6ʗ3]*[20ʗ4]/[26ʗ7]

=0.1473092125

3) PQX

Where P = 0.35

q = 1- P = 1- 0.35 = 0.65

x = 7

= 0.35 * 0.657

= 0.35 * 0.0490222789062

= 0.0171577976171

X!

Where ƛ =50/12 =4

e = 2.7183

X= 0

40 x 2.7183-4 1 x 0.0183 = 0.0183

0! 1

Where ƛ =4

e = 2.7183

X= 7

47 x 2.7183-4 16384 x 0.0183 = 299.8272

7! 5040 5040

= 0.0595

OSSAI MARY AMARACHI

2019/243684

ECONOMICS

Amxrachukwu@gmail.com

1a Poisson probability distribution :

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates.

The Poisson Distribution is: P(x; μ) = (e-μ * μx) / x!

Where:

The symbol “!” is a factorial.

μ (the expected number of occurrences) is sometimes written as λ. Sometimes called the event rate or rate parameter.

b.Uniform probability distribution:In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

Px=n/1

where:Px=Probability of a discrete value

n=Number of values in the range

c. Exponential probability distribution:

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. the exponential distribution is the probability distribution of the time between events in a poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

2. probability of picking just one doctor =6/26

Probability of picking just one nurse=20/26

Therefore the probability of picking 4 doctors and 6nurses=

4 doctors=4*6/26= 24/26= 12/13

6 nurses=6*20/26=120/26=60/13

3.This implies that failure occur 7 times before she got married in her 8th relationship therefore, let x be the number of her relationships before her marriage the probability function is P(X=x)

= p(1-p)(_^x) or pq^x.

P=0.35 and q=1-0.35= 0.65

P(x=7)= 0.35*(0.65)〖^7〗.=0.01716

4. P(x; μ) = (e-μ * μx) / x!

x is the number of chocolate chips

to get lambda,50/12=4

e=2.71828

a) no chocolate chip

x=0

p(x=0)=(e^(-4).4^o)/0!=0.018321/1=0.018321.

b) 7 chocolate chips

x=7

p(x=7)=((e^(-4).4^7 ))/7!=(0.01832*16384)/5040

= 0.05955.

OSSAI MARY AMARACHI

2019/243684

ECONOMICS

Amxrachukwu@gmail.com

1a Poisson probability distribution :

The Poisson distribution is a discrete distribution that measures the probability of a given number of events happening in a specified time period. In finance, the Poisson distribution could be used to model the arrival of new buy or sell orders entered into the market or the expected arrival of orders at specified trading venues or dark pools. In these cases, the Poisson distribution is used to provide expectations surrounding confidence bounds around the expected order arrival rates.

The Poisson Distribution is: P(x; μ) = (e-μ * μx) / x!

Where:

The symbol “!” is a factorial.

μ (the expected number of occurrences) is sometimes written as λ. Sometimes called the event rate or rate parameter.

b.Uniform probability distribution:In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

Px=n/1

where:Px=Probability of a discrete value

n=Number of values in the range

c. Exponential probability distribution:

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. the exponential distribution is the probability distribution of the time between events in a poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

2. probability of picking just one doctor =6/26

Probability of picking just one nurse=20/26

Therefore the probability of picking 4 doctors and 6nurses=

4 doctors=4*6/26= 24/26= 12/13

6 nurses=6*20/26=120/26=60/13

3.This implies that failure occur 7 times before she got married in her 8th relationship therefore, let x be the number of her relationships before her marriage the probability function is P(X=x)

= p(1-p)(_^x) or pq^x.

P=0.35 and q=1-0.35= 0.65

P(x=7)= 0.35*(0.65)〖^7〗.=0.01716

4. P(x; μ) = (e-μ * μx) / x!

x is the number of chocolate chips

to get lambda,50/12=4

e=2.71828

a) no chocolate chip

x=0

p(x=0)=(e^(-4).4^o)/0!=0.018321/1=0.018321.

b) 7 chocolate chips

x=7

p(x=7)=((e^(-4).4^7 ))/7!=(0.01832*16384)/5040

= 0.05955.

1a. Poisson probability distribution:it is used to show how many times an event is likely to occur over a specified period.they are often used to understand independent events that occurs at a constant rate within a given interval of time.it is a discrete function, ie the variable can only take specific values in a list.

B. Uniform probability distribution: are probability distribution with equal likely outcome.it is of two types namely: discrete and continuous whose outcomes are the same and infinite, respectively.

C.Exponential probability distribution: is often concerned with the amount of time until some specific events occurs.

2. The probability =0.2206

3. The probability =0.01712

4. P(X=0) 0.0183

B. P(X=7)=0.0595

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value in a list Examples include number of mosquito bite on person, number of computer crashes in a day etc.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda to the nearest whole number. What is the probability that it contains?

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda to the nearest whole number. What is the probability that it contains?

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

To make a dozen chocolate chips cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a chocolate is chosen at random from a batch, also using lamda to the nearest whole number. What is the probability that it contains?

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value in a list Examples include number of mosquito bite on person, number of computer crashes in a day etc.Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

Poisson probability distribution: it is used to show how many times an event is likely to occur over a specified period. It is a discrete function by which variable can take only a specified value in a list Examples include number of mosquito bite on person, number of computer crashes in a day etc.Uniform probability distribution: it is also known as Continuous uniform distribution or rectangular distribution. It is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.

Exponential probability distribution: it is a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the probability distribution of the time between events in a Poisson point process.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

ECO231 ASSIGNMENT.

Write short notes on the following types of probability distributions:

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

Uzoka ikechukwu precious

2029/249450

1a) Poisson Probability Distribution

In probability theory and statistics, the Poisson distribution French pronunciation: named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.[1] The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

b) Uniform Probability Distribution

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

c) Exponential Probability Distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 × 0.65^7

= 0.0176.

a.No chocolate

b.7 chocolates.

a. P(x=0) e^-50×50^0/0!

=0.0067×1/1

=0.0067

b. P(x=7)e^-50 ×50^7/7!

= 0.0067×78125/5040

=523.4/5040

=0.1038.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

P {X=4 doctors and 3 nurses}

N=6+20=26

n=7

r=6

x= 4 doctors 3 nurses

P{X=x} =[rCx][N-rCn-x]/NCn

=(6C4) (20C3)/26C7

= 15×1140/657800

= 17100/657800

= 0.026.

P(X=x) Pqx

P= 0.35

q= 1-p = 1-0.35= 0.65

x = x-1 = 8-1= 7.

P(x=7) = 0.35 ×

= 0.0176.

a.)No chocolate

b.)7 chocolates.

Solution:

50/12=4.1667~4

a.)π^x.e^-π/x!

P(x=0) =4^0×2.7183^-4/0!

=0.0183×1/1

=0.0183.

b.)π^x.e^-π/x!

P(x=7) =4^7×2.7183^-4/5040

= 16384×0.0183/5040

=300.07/5040

=0.0595.

Answers

a. Poission probability distribution: it is a probability distribution that is used to show how many times an event is likely to occur over a specific period of time. It is used in testing how unusual an event frequency is for a given interval.

It is a discrete function, meaning that the variable can only take specific values in a list. Examples: i. The number of computer crashes in a day.

ii. Then numbers of mosquito bites on a person. e.t.c.

b. Uniform probability distribution: lt is a continuous probability distribution and is concerned with events that are equally likely to occur. It is constant since each variable has equal chances of being the outcome.

c. Exponential probability distribution: lt is often concerned with the amount of time until some specific event occurs. It is usually present when we are dealing with events that are rapidly changing early on for. Examples: News, blog, length, in minutes, of long distance business telephone calls. e.t.c.

2. Hypergeometric distribution:

N=26, n=7, r=6

6C4 × 20C3/26C7

=15 × 114O/657800

=0.026

3. Geometric distribution:

P=0.35

q= 1-0.35 =0.65

=0.35 ×(0.67)^7

=0.35× 0.049

=0.01715

4. Poission distribution

e=2.7183

Lambda= 50

a. 2.7183 ^-50 × 50^0/0!

=5.185

b. 2.7183^-50 × 50^7/ 7!

=8.0368

Name:Ahamefula Miracle Chisom

Reg no: 2017/249478

Dept:Economics

*ECO 231 ASSIGNMENT*

1. Write short notes on the following types of probability distributions:

a] Poisson probability distribution

b] Uniform probability distribution

c] Exponential probability distribution

2] Six(6) doctors and twenty(20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are randomly selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

3] Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35. Find the probability that she will actually get married in her 8th relationship.

4} To make a dozen chocolate cookies, 50 chocolate chips are mixed into the dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using Lambda (the mean) to the nearest whole number. What is the probability that it contains:

a] No chocolate chips?

b} 7 chocolate chips?

What Is a Poisson probability Distribution?In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

What Is Uniform probability Distribution?In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

Understanding Uniform probability Distribution

There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. There are only 6 possible values to return and nothing in between.Some uniform distributions are continuous rather than discrete. An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there is an infinite number of points between 0.0 and 1.0.

Exponential distributionIn probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

(somewhat formal) definition for the hypergeometric distribution, where X is a random variable, is:

hypergeometric distribution formula(somewhat formal) definition for the hypergeometric distribution, where X is a random variable, is:

hypergeometric distribution formula

Where:

K is the number of successes in the population

k is the number of observed successes

N is the population size

n is the number of draws

Solution

Formula

P(x=k) =(Kck) (N-Kcn-k)/Ncn

2. 6c4*20c3/26c7 =15*1140/657800

=0.0260

Formula

P(x=8)=(1-p)^x-1*p

3. P(x=8)= (1-0.35)^8-1*0.35

P(x=8) =(1-0.35)^7*0.35

=0.0490*0.35

=0.01715

Formula

e^-∆*∆^x/x!

4I. No chocolate Chip

∆=50/12 =4.167

X=0

P(x= 0) e^-4.167(4.167)^0/0!

P(x=0) =0.0155*1/0! =0.0155

4ii. 7 chocolate Chip

P(x=7) e^-4.167(4.167)^7/7!

P(x=7) e^-4.167*21815.46/7*6*5*4*3*2*1

=0.0155*21875.46/7*6*5*4*3*2*1

=0.0155*21875.46/5040

=339.07/5040

=0.067276

Ugwuanyi Nkeonye Laurel

2019/243315

nkemlaurel@gmail.com

1. Poisson distribution

In statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period.

A Poisson distribution can be used to estimate how likely it is that something will happen “X” number of times. For example, if the average number of people who buy cheeseburgers from a fast-food chain on a Friday night at a single restaurant location is 200, a Poisson distribution can answer questions such as, “What is the probability that more than 300 people will buy burgers?”

e is Euler’s number (e = 2.71828…)

x is the number of occurrences

x! is the factorial of x

λ is equal to the expected value (EV) of x when that is also equal to its variance

It can be used to estimate how many times an event is likely to occur within “X” periods of time.

Poisson distributions are used when the variable of interest is a discrete count variable.

Many economic and financial data appear as count variables, such as how many times a person becomes unemployed in a given year, thus lending themselves to analysis with a Poisson distribution.

Uniform distributionThe uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.

A uniform distribution, also called a rectangular distribution, is a probability distribution that has constant probability.

This distribution is defined by two parameters, a and b:

a is the minimum.

b is the maximum.

The distribution is written as U(a, b).

The probability density function is

F(x)= 1/b-a

for a ≤ x ≤ b.

Uniform distributions are probability distributions with equally likely outcomes.

In a discrete uniform distribution, outcomes are discrete and have the same probability.

In a continuous uniform distribution, outcomes are continuous and infinite.

In a normal distribution, data around the mean occur more frequently.

The frequency of occurrence decreases the farther you are from the mean in a normal distribution.

Exponential distributionThe exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. We will now mathematically define the exponential distribution, and derive its mean and expected value. Then we will develop the intuition for the distribution and discuss several interesting properties that it has.

A continuous random variable X is said to have an exponential distribution with parameter λ>0, shown as X∼Exponential(λ),

2. P(x-r) =. (rCx * N-rCn-x)/ NCn

N= 6+20 =26

r=6

n=7

X =4

P(X=4)= (6C4 × 26-6C7-4)\ 26C7

P(X=4) = (6C4 × 20C3)/26C7

P(X=4)= 0.0024995743

~0.0025

3. P(X=X) = owe

P= 0.35

q= 1-0.35 = 0.65

P(X=7)= 0.35×0.65^7

0.35×0.0490

= 0.0172

4. P(x; μ) = (e^-μ) (μ^x) / x!

e= 2.7183

μ= 50/12 = 4.167

I. P(X=0) =

(2.7183^-4.167)(4.167^0)/0!

P(X=0) = 0.015498

~0.0155

ii. P(X=7) =

(2.7183^-4.167)(4.167^7)/7!

(0.015498×21815.4664)/5040

=0.06708

~0.067

NAME: ATTAMA LILLIAN OGECHUKWUIn statistics, a Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period. In other words, it is a count distribution. Poisson distributions are often used to understand independent events that occur at a constant rate within a given interval of time. It was named after French mathematician Siméon Denis Poisson.

The Poisson distribution is a discrete function, meaning that the variable can only take specific values in a (potentially infinite) list. Put differently, the variable cannot take all values in any continuous range. For the Poisson distribution (a discrete distribution), the variable can only take the values 0, 1, 2, 3, etc., with no fractions or decimals.

Six (6) doctors and twenty (20) nurses attend a small conference. All 26 names are put in a hat, and 7 names are selected without replacement. What is the probability that 4 doctors and 3 nurses are picked?

REG NO: 2019/243411

Write short notes on:

Poisson Distribution

Poisson distribution can be used to estimate how likely it is that something will happen “X” number of times. For example, if the average number of people who buy cheeseburgers from a fast-food chain on a Friday night at a single restaurant location is 200, a Poisson distribution can answer questions such as, “What is the probability that more than 300 people will buy burgers?” The application of the Poisson distribution thereby enables managers to introduce optimal scheduling systems that would not work with, say, a normal distribution.

Uniform Distribution

A uniform probability distributions one in which all elementary events in the sample space have an equal opportunity of occurring that is each event occurring has an equal chance. As a result, for a finite sample space of size n, the probability of an elementary event occurring is 1/n.

Exponential Distribution

This is usually present when we are dealing with events that are rapidly changing early on for eg news and blog. The exponential distributions wildly used in the field of reliability. Reliability deals with the amount of time a product lasts. Its probability function is represented as; P(X=x) = me-mx.

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

No chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=0) = e-4(4)0/0! = 0.0183

7 chocolate chips

Solution

P(X=x) = e-ƛƛx/x!

ƛ = 50/12 = 4 (to the nearest whole number)

P(X=7) = e-4(4)7/7! = 0.0595

Name: Onyeleonu precious Oluomachi

Reg no: 2019/248162

Write short notes on:

POISSON PROBABILITY DISTRIBUTION:

This is a discrete probability distribution derived by a French Mathematician/Physicist, SIMEON POISSON in 1837..

It arises frequently in counting number of random events in a given interval of time of space in a particular area of opportunity..

WHEN TO APPLY POISSON DISTRIBUTION

a) When we want to determine the number of times and event occured in a given area of opportunity..

b)When the number of times and event occured in one area is independent of the number of times and event occured in another area of opportunity.

c)When the probability of events occuring in an area becomes zero as the area of occurance becomes smaller..

POISSON PROBABILITY FUNCTION: e-^ ^X/x!

-Where x=random variables

-The symbol e(exponential)(the base of the natural logritgm system)is a constant =2.71282

-The Greek letter Lamb is called the parameter of the distribution, represents average number of times the random variable occurs in a specified interval of time or space..

UNIFORM PROBABILITY DISTRIBUTION

This is a type of continuous distribution which is used in rounding errors when measurements are recorded to some degree of accuracy. The parameters a and B which form the two boundaries of the interval over which the distribution is defined are the two parameters that completely specified the distribution..

The density function forms a rectangle with base B-a and constant height (B-a),as a result,the uniform distribution is often called RECTANGULAR DISTRIBUTION.The application of this distribution is often based on assumption that the probability of falling in an interval of fixed length with (a-B) is constant.

This distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. It is often abbreviated as U(a-B), where U stands for uniform distribution..

The difference between the bounds defines the interval lenght. All intervals of the same length on the distribution’s support,are equally probable..

Exponential Distribution

This is usually present when we are dealing with events that are rapidly changing early on for eg news and blog. The exponential distributions wildly used in the field of reliability. Reliability deals with the amount of time a product lasts. Its probability function is represented as; P(X=x) = me-mx.

Solution3. Using the geometric approach, suppose that the probability of female celebrity getting married is 0.35, find the probability that she wll actually get married in her 8th relationship.

P(X=x)= r N-r

x n-x

N

n

= N = 6+20 = 26

r = 6

n = 7

x = 4

P(X=4) = 6 26-6

4 7-4

26

7

P(X=4) = 6C4 X 20C3

26C7

P(X=4) = 15X1140/657800

P(X=4) = 17100/657800

= 0.0025995743

= 0.0260. 4dp

Solution

P(X=x)=Pqx

P= 0.35

q= 1-p, 1-0.35= 0.65

x= 8-1= 7 (number of failures)

P(X=7) = 0.35x(0.65)7

= 0.017157

= 0.017157×100

= 1.7% chance.

4. To make a dozen chocolate cookies, 50 chocolate chips are mixed into a dough. The same proportion is used for all batches. If a cookie is chosen at random from a large batch, also using landa (the mean) to the nearest whole number.