Eco. 231–Economic Statistics (Online Discussion/Quiz 1-Probability distributions:—17-11-2021)

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 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.

(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

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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 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.

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

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.

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.

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.

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.
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.
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.

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.

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 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 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: 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

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; 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: 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

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: 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: 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.

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 (final answer)
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.(final answer)
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(final answer)

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

Arinze ebuka Kelvin
2019/246530
Economics major
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.
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

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 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

Daniel Unique Agbenu
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 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.
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

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).

=0.0183156388
≈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).

=0.0183156388
≈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

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

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.

Name: Nwodo Micheal Chidera
Reg No: 2019/243281
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.

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.

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

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

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 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: 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

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

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

Ok. Okoro David KOSISOCHUKWU 2019/241946

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

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

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

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

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: 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

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

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

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

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

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

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 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.

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) 2.7183-⁷×5⁷/5040
= 0.0067×78125/5040
=523.4/5040
=0.1038

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 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.

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
£= 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

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

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

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.

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.

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) 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

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

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

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

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

ASSIGNMENT ON ECO 231 Date: 23/11/2021
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
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
(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

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:. 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

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

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.

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.

NAME: Nebo Casmir Chukwuemeka

REG. NO: 2019/244263

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