Basics of Probability Theory, Random Variables, Discrete and Continuous, moments
and other functions of Random Variables, Limit Theorems and Inequalities, Poisson
Process, Markov Chains.

Probability Theory and Applications.
Course objective
To familiarise students with basic concepts of Probability Theory with reasonable amount
of rigor. Examples will be used for motivating students to learn and understand the basic
concepts.

S. No.

Lectures

No. of Lectures

1.

Combinatorial Analysis.
Basic principle of counting, permutation and combinations, multinomial coefficients.

2

2.

Axioms of Probability.
Sample Space and Events, Sigma Algebra of Events, Axioms of Probability, Some
simple Propositions, Conditional Probability, Baye's Rule, Independent Events,
Additional Properties. Problems.

3

3.

Random Variables
Definitions and Examples, Discrete Random Variables, Probability
Mass Function and Distribution Function, Special kinds of Discrete Random Variables.
Continuous Random Variables, Probability Density Function and Distribution Function,
Properties of Distribution Function, Special kinds of Continuous Random Variables,
Normal Approximation of Binomial and Poisson Random Variables, Continuity
Correction, Computer Generation of Random Variables. Problems.

10

4.

Jointly Distributed Random Variables.
Joint Distribution Functions, Independent Random Variables, Sums of Independent
Random Variables, Conditional Distribution, Discrete and Continuous Cases, Order
Statistics, Joint Probability Distribution Function and Marginal Probability Distribution
Function, Multivariate Normal Numbers. Problems.

4

5.

Moments.
Expectation of Random Variables, Expectation of Sums of Random Variables,
Expectation of Special Discrete Random Variables, Expectation of Special Continuous
Random Variables, Variance, Conditional Expectation, Computing Expectation by 4
Conditioning, Moment Generating Function, Characteristic Function. Problems.

8

6.

Equalities and limit laws.
Markov and Chebychov’s Inequalities, Convergence Concepts, Law of Large Numbers,
Central Limit Theorem and its Applications, Other Inequalities. Problems.

8

7.

Applications.
The Poisson Process, Markov Chains.

5

Total

40

Class XII Differential and Integral Calculus.

A first Course in Probability, Sheldon Ross, 9^{th} Edition, 2012, PEARSON.

An Introduction to Probability Theory and its Applications. Feller, Volume I & II.

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