The document presents an introduction to probability and various discrete distributions, detailing concepts such as random experiments, sample spaces, probability spaces, and the axioms of probability. It covers special distributions including uniform, Bernoulli, binomial, and Poisson, along with their probability mass functions, means, and variances. The author expresses gratitude to mentors and outlines plans for further study of more distributions and random variables.

1. ACCADEMIC WRITING
TOPIC:INTRODUCTION TO PROBABILITY AND
SOME SPECIAL DISCRETE DISTRIBUTIONS
SUBMYTTED BY
DOYEL GHOSH
MAIL:doyel11111996@gmail.com

2. ACKNOWLEDGEMENT
 I would like to express my special thanks of gratitude to
Swayam who gave this wonderful opportunity to take a
presentation on the topic Introduction to Probability and
Some Discrete Distributions as well as professor Dr. Ajay
Semalty who offer the course Academic Writing.

3. Contents:
 Random Experiment
 Sample Space
 Probability Space
 Probability Axioms
 Random Variable
 Uniform Distribution
 Bernoulli Distribution
 Binomial Distribution
 Poisson Distribution

4. Random Experiment
 A random experiment is an experiment in which:
 All outcomes of the experiment are known in advance.
 Any performance of the experiment results in an outcome that is not known in
advance.
 The experiment can be repeated under identical conditions.

5. Sample Space
The sample space of a statistical experiment is a pair (Ω,Ş) where
 Ω is the set of all possible outcomes of experiment.
 Ş is σ-field of subsets of Ω.
Note:
1. Events: Subsets of Ş .
1. Ω can be countably finite , countably infinite or uncountably many.

6. Probability Space
 The triple (Ω,Ş,P) is called a probability space.
 Note:
 Ş is the σ-algebra over Ω and is the collection of events which are subset of the
power set of Ω such that
1. φ belongs to Ş.
2. If A1 ,A2 ,…,An belongs to Ş then UAi belongs to Ş.
3. A belongs to Ş implies A’ belongs to Ş.
 Example:
If X = {a, b, c, d}, one possible σ-algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }.

7. Probability Axioms
Let P be the set function such that
 P(A) ≥ 0 for all A belonging to Ş.
 P(Ω)=1.
 If A1 , A2 ,…, An are mutually exclusive then P(UAi)=∑P(Ai).
P is called probability measure.
Example:
Suppose we have three numbers {1,2,3} with probability measure ½ , ¼ and ¼.
Then P(1),P(2),P(3) lies in the interval [0,1] and P{1,2,3}=1 again UP(n) = ∑ P(n)
,where n=1,2.

8. Random Variable
A finite single valued function X that maps Ω into R is called random variable if the
inverse image under X of all Borel sets in R are events i.e if
X-1(B)={ω : X(ω) belongs to B} belongs to Ş for all B belongs to ß.
Result:
X is a random variable iff for each x belonging to R the set
{ω: X(ω) ≤ x} belongs to Ş.
Note:
 A random variable X is said to be discrete if it can take countable number of
values within its range.
 A random variable X is said to be continuous if it can take uncountably infinite
values within its range.

9. Uniform Distribution
X is said to have a uniform distribution on n points {x1 ,x2 ,…,xn} if its pmf is of the form
ƒ(xi)=1/n , i=1,2,…,n
 Mean of Uniform distribution:
E(X)=(n+1)/2 (in particular when xi = i for all i)
 Variance of Uniform Distribution:
V(X)=(n+1)/16 (in particular when xi = i for all i)
 Example:
A die roll has four possible outcomes: 1,2,3,4,5, or 6. There is a 1/6 probability
for each number being rolled.

10. Bernoulli Distribution
The Assumptions of Bernoulli Trials:
 Each trial results in one of two possible outcomes, denoted success or failure.
 The probability of success remains constant from trial-to-trial and is denoted by
p. Write q = 1−p for the constant probability of failure.
 The trials are independent.

11. PMF of Bernoulli Distribution
ƒ(x)= px q(1-x) , where x=0,1
 Mean of Bernoulli Distribution:
E(X)=p
 Variance of Bernoulli Distribution:
V(X)=pq
 Example:
Births: How many boys are born and how many girls are born each day.
Here we can take the birth of a boy to be failure and the birth of a girl to be success.

12. Binomial Distribution
Suppose that there are n Bernoulli trials. Let X denote the total number of successes
in the n trials. The probability distribution of X is given by:
ƒ(x)=nCx px q(n-x) , where x=0,1,2,…,n
 Mean of Binomial Distribution:
E(X)= np
 Variance of Binomial Distribution:
V(x)= npq
 Example:
A fair die is rolled n times. The probability of obtaining exactly one 6 is n(1/6)(5/6)n-1 .

13. Poisson Distribution
A random variable X is said to follow Poisson Distribution if its pmf is given by
ƒ(x)=exp(-λ) λx/x! x=0,1,...
 Mean of Poisson Distribution:
E(X)=λ
 Variance of Poisson Distribution:
V(X)=λ

14. Binomial Distribution converges to
Poisson Distribution
Conditions:
 Number of trials is sufficiently large.
 Probability of success is very small.
Under these conditions Binomial distribution can be approximated by poisson
distribution with parameter λ=np.

15. Future Planning
 I have discussed here about the axiomatic approach of probability theory and
some discrete distribution. Further I will study about more discrete and
continuous distributions and form a new distribution followed by some random
variables.

16. THANK YOU.