I HAVE DISCUSSED THEORY OF PROBABILITY IN AXIOMATIC APPROACH AS WE KNOW THERE WERE MANY DRAWBACKS OF CLASSICAL APPROACH. I HAVE ALSO DISCUSSED ABOUT SOME DISCRETE PROBABILITY DISTRIBUTIONS.

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.