This document provides an assignment on discrete probability distributions, including binomial, Poisson, and negative binomial distributions. It defines each distribution and provides examples of their properties and applications. It also includes numerical problems demonstrating how to fit data to each distribution and calculate relevant probabilities.

1. ACHARYA NARENDRA DEVA UNIVERSITY OF AGRICULTURE OF TECHNOLOGY,
KUMARGANJ, AYODHYA-(224229), U.P.
Assignment
on
Discrete probability distributions
Course No : STAT-502 4(3+1)
Course name : Statistical methods for applied sciences
Presented to:
Dr. Vishal Mehta
Assistant Professor
Department of Agril. Statistics
Presented by:
Dharmendra Kumar
Id. No. A-12993/22
Ph.D. 𝟏𝒔𝒕
𝐒𝐞𝐦𝐞𝐬𝐭𝐞𝐫
Soil Science and Agril. Chemistry

2. Content
• Discrete probability distributions
• Types of discrete probability
distribution
• Binomial distribution
• Poisson distribution
• Negative Binomial distribution
• Numerical
• References

3. Discrete probability distributions
A discrete probability distributions counts
occurrences that have countable or finite
outcomes. This is contrast to a continuous
distribution, where outcomes can fall anywhere on
a continuum.
 Type of discrete probability distributions-
 Binomial distribution
 Poisson distribution
 Negative binomial distribution

4. Binomial distribution
• Binomial distribution given by James
Bernoulli.
• It is discrete type of probability distribution.
• It is extended or generalization version of
Bernoulli distribution.
• It is arise when Bernoulli trails are performed
repeatedly for a fixed number of times say ̔ n ̕ .

5. Definition:
A random variable ̔ x ̕ is said to follow
Binomial distribution, if it assumes non –
negative values and its probability mass
function is given by :-
P(X=x) = { 𝑛
𝑥
𝑝𝑥
𝑞𝑛−𝑥
x= 0,1,2,3,……,n

6. • The two independent constant ̔ n ̕ and ̔ p ̕ in
the binomial distribution are known as the
parameters of the distribution.
Where,
n = Number of trails
p = Probability of success
q = Probability of failure (n-x)
x = Number of success

7. Assumptions of binomial distribution
• The number of trails ̔ n ̕ is finite.
• The probability of success ̔ p ̕ is constant
for each trail.
• The trails are independent of each other.
• Each trail must result in only two
outcome i.e; success or failure.

8. Properties of binomial distribution
1. Mean(np) > Variance(npq)
2. Standard deviation = √npq
3. Coefficient of skewness(β1) =
𝑝−𝑞
√𝑛𝑝𝑞
4. Coefficient of kurtosis(β2) =
1−6𝑝𝑞
𝑛𝑝𝑞

9. 4. Characteristic function
(Φ𝑥(t)) =(p𝑒𝑖𝑡
+q)
𝑛
5. Moment generating function
(𝑀𝑥(t)) =(p𝑒𝑡
+q)
𝑛
6. Probability generating function
(𝑍𝑥(t))=(𝑍𝑝 + 𝑞)𝑛
7. Cumulant generating function
(𝐾𝑥(t)) =log𝑀𝑥(t)
Properties of binomial distribution

10. Applications of Binomial distribution
1. Take positive or negative reviews on any
product from public.
2. Using yes or no survey in an event.
3. To know the plat diseases occurrence or
not occurrence among plants.
4. In quality control, officer may want to
know & classify items as defective or
non-defective.

11. Numerical problem
Question: 7 coins are tossed and number of head
noted. The experiment is repeated 128 times and
the following distribution is obtained. Fit in
binomial distribution.When the coin is unbiased.
Solution:
When the coin is unbiased, then
p=q=1/2, p/q=1
n = 7 and N = 128
No. of heads 0 1 2 3 4 5 6 7 Total
Frequencies 7 6 19 35 30 23 7 1 128

12. X f fx n-x/x+1 p(x)=n-x/x+1 × p/q F(x)
0 7 0 7 7 f(0)=Np(0)= 1
1 6 6 3 3 f(1)=1×7= 7
2 19 38 5/3 5/3 f(2)=7×3= 21
3 35 105 1 1 f(3)=21×5/3= 35
4 30 120 3/5 3/5 f(4)=35×1= 35
5 23 115 1/3 1/3 f(5)=35×3/5= 21
6 7 42 1/7 1/7 f(6)=21×1/3= 7
7 1 7 f(7)=7×1/7= 1
Total N=128 Σ f(x) 128
p(0)= 7C0 p0 q7-0 = 7C0 (1/2)0 (1/2)7 = 1/128
So that f(0) =Nqn =128(1/2)7 = 1
Fiting in binomial distribution

13. Poisson distribution
The Poisson distribution named Simon
Denish Poisson.
It describe random events that occur
rarely over a limit of time and space.
It is expected in cases where the chance
or probability of any individual events
being success is very less.
Poisson distribution is a limiting case of
binomial distribution.

14. Definition:
If x is a Poisson variate with
parameter λ =np, then the probability that
exactly x events will occur in a given by
probability mass function as :
p(x)= {
𝑒−λλ𝑥
𝑥!
; 𝑥 = 0,1, … .
Where, λ is known as parameter of the
distribution so that λ > 0
x= Poisson variate , e= 2.7183

15. Constant of Poisson distribution
• Mean= λ
• Variance= λ
• Mean=Variance= λ
• Standard deviation= λ
• Coefficient of skewness(β1 ) =
1
λ
• Coefficient of kurtosis (β2) = 3+
1
λ

16. Constant of Poisson distribution
• γ1= β1
• γ2= β1-
1
3
• Moment generating function-
(𝑀𝑥(t))=𝑒λ(𝑒𝑡−1)
• Probability generating function-
(𝑍𝑥(t))=𝑒λ(𝑧−1)
• Characteristic function-
(Φ𝑥(t)) = 𝑒λ(𝑒𝑖𝑡−1)

17. Examples of Poisson distribution
• The number of blinds born in a town in a
particular year.
• The number of mistakes committed in a
typed page.
• The number of students scoring very high
marks in all subject.

18. Numerical problem
Question: Fit a Poisson distribution to the following data.
Solution:
N= 500
Σfx= 986
Mean=λ =1/N× Σfx
λ = 1/500 × 986
λ = 1.972
x 0 1 2 3 4 5 6 7 8
f 54 156 132 92 37 22 4 0 1

19. x f xifi λ/x+1 p(x) f(x)=Np(x)
0 56 0 1.972 0.13920 69.6000 − 70
1 156 156 0.986 0.27455 137.2512 − 137
2 132 264 0.657 0.27006 135.3296 − 135
3 92 276 0.493 0.17793 88.9566 − 𝟖𝟗
4 37 148 0.394 0.10964 43.8556 − 44
5 22 110 0.328 0.03459 17.2966 − 𝟏𝟕
6 4 24 0.281 0.01137 5.6846 − 6
7 0 0 0.247 0.00320 1.6013 − 2
8 1 8 0.219 0.00078 0.3942 − 𝟎
Total N=500 𝜮𝒙𝒊𝒇𝒊 = 𝟗𝟖𝟔
p(x)= {
𝑒−λλ𝑥
𝑥!
p(0)= {
𝑒−1.9721.9720
𝑥!
= 𝒆−𝟏.𝟗𝟕𝟐

20. Negative Binomial distribution
• Negative Binomial distribution is given
by Blaise Pascal (1679).
• In Negative Binomial distribution the
number of success is fixed and number of
trails is a independent.
• Negative Binomial distribution is also
known as Pascal`s distribution.

21. Definition:
A random variable x is said to
follow a negative binomial distribution
with parameters r and p, if its probability
mass function is given by:
P(X=x)= { 𝑥+𝑟−1
𝑟−1
𝑝𝑟
𝑞𝑥
; x= 0,1,2,……∞
Where,
x = Number of trails
r = Number of success
p = Probability of success
q = Probability of failure

22. Properties of Negative Binomial distribution
• If r=1, Negative Binomial distribution
tends to Geometric distribution.
• If r =∞, q=0 and rq=λ, then Negative
Binomial distribution tends to Poisson
distribution.
• Variance(rpq) > Mean(rp)
• The distribution is positively skewed and
leptokurtic.

23. Properties of negative binomial distribution
• Moment generating function-
(𝑀𝑥(t)) =(Q−P𝑒𝑡
)
−𝑟
• Coefficient of skewness(β1 ) =
𝑄+𝑃
r𝑃 𝑄
• Coefficient of kurtosis (β2) = 3+
1+6𝑃𝑄
rPQ
• γ1=
𝑄+𝑃
𝑟𝑃𝑄
• γ2=
1+6𝑃𝑄
rPQ

24. Application of Negative Binomial distribution
• It is applicable in those data set where
variance is greater than mean.
• When Poisson distribution unable to
describe a data set or inadequate then we
prefer negative binomial distribution.
• Used in accident statistics(Birth and
Death).
• Used in psychological data set.

25. Numerical problem
Question: Fit a Negative Binomial distribution.
Solution:
Let x be the number of demands per day
Here x ~ NB(K,P)
The p.m.f of NB (K,P) is
P(X=x)= { 𝑥+𝐾−1
𝑟−1
𝑝𝐾
𝑞𝑥
; x= 0,1,2,……∞
0<P<1, If q=1-p
K>0
x 0 1 2 3 4 5 6
f 47 50 25 14 10 6 0

26. x f fx 𝒇𝒙𝟐 𝒙+𝑲
𝒙+𝟏
× 𝒒 p(x) Expected frequency=Np(x)
0 47 0 0 0.9961 0.3104 46.56
1 50 50 50 0.6389 0.3092 46.3767
2 23 46 92 0.5198 0.1975 29.6313
3 14 42 126 0.4603 0.1027 15.3997
4 10 40 160 0.4245 0.0473 7.0908
5 6 30 150 0.4245 0.241 3.619
6 0 0 0 0 0.0088 1.32
Total N=150 𝜮𝒙𝒇 = 𝟐𝟎𝟖 𝜮𝒇𝒙𝟐
= 𝟓𝟕𝟖
150
Mean =
𝜮𝒇𝒙
𝜮𝒇
=
𝟐𝟎𝟖
𝟏𝟓𝟎
= 𝟏. 𝟑𝟖𝟔𝟕
Variance =
𝜮𝒇𝒙𝟐
𝜮𝒙𝒇
-(𝑥)2
=
𝟓𝟕𝟖
𝟐𝟎𝟖
−(1.3867)2
= 1.9309
𝑝=
𝒎𝒆𝒂𝒏
𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆
=
𝟏.𝟑𝟖𝟔𝟕
𝟏.𝟗𝟑𝟎𝟗
= 0.7183, 𝑞 = 1- 𝑝 = 1- 0.7183= 0.2817
𝑘= 𝑥 ×
𝑝
𝑞
= 1.3867 ×
𝟎.𝟕𝟏𝟖𝟑
𝟎.𝟐𝟖𝟏𝟕
= 3.5359

27. p(x=0)= 𝑝𝑘
= (0.7183)3.5359
= 𝟎. 𝟑𝟏𝟎𝟒
p(x=1)= (𝒙+𝑲
𝒙+𝟏
× 𝒒) × p(x=0)
= k.p.q(x=0)
= 3.5359 × 0.2817 × 0.3104
= 0.3092
p(x=2)= (𝒙+𝑲
𝒙+𝟏
× 𝒒) × p(x=1) = 0.1975
p(x=3)= 0.1027
p(x=4)= 0.0473
p(x=5)= 0.241
p(x=6)= 0.0088

28. References
• Agarwal, B.L., Programmed Statistics.
New Age International Publishers, New
Delhi, 3rd edition.
• Gupta, S.C. and Kapoor, V.K.,
Fundamentals of Mathematical Statistics.
Sultan Chand & Sons, New Delhi, 12th
edition.
• Paul, N.C., Statistics in shorts. New
Vishal Publications, New Delhi.