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This is an introductory course on probability theory.
The aim of the course is to give the students a working knowledge of
probability and there will be an emphasis on solving problems.
I plan to cover the following topics, some topics maybe dropped/added depending on the pace of the course.
More details can be found at the
course webpage
https://courses.iitm.ac.in

References:

Probability and Random Processes with Applications to Signal Processing, 3rd ed, H. Stark and J. W. Woods.

Probability, Random Variables, and Stochastic Processes, 4th ed., A. Papoulis and S. U. Pillai

Course topics.

- Introduction
- Review of set theory
- Axiomatic probability
- Discrete random variables
- Continuous/mixed random variables
- Transformation/functions of random variables
- Operations on random variables
- Generating functions of random variables
- Inequalities
- Asymptotic results

Grading policy (Tentative)

Quiz I : 25%, Quiz II :25%, Endsem : 50%

Exams

- Quiz I : 20 Feb 2016

- Quiz II : 02 Apr 2016

- Endsem : 03 May 2016

Lectures

- 11 Jan Lecture 1: Introduction: three views of probability

- 12 Jan Lecture 2: Review of set theory

- 13 Jan Lecture 3: Probability terminology, sigma algebras

- 14 Jan Lecture 4: Axioms of probability, probability spaces

- 18 Jan Lecture 5: Probability spaces, conditional probability

- 19 Jan Lecture 6: Conditional probability

- 20 Jan Lecture 7: Independence

- 21 Jan Lecture 8: Product spaces

- 27 Jan No class

- 28 Jan Lecture 9: Combinatorics, urn and occupancy problems

- 29 Jan Lecture 10: Application to communication (MAP decoder)

- 01 Feb Lecture 11: Discrete random variables

- 02 Feb No class

- 03 Feb Tutorial-1

- 04 Feb Lecture 12: Discrete random variables

- 08 Feb Lecture 13: Pairs of random variables

- 09 Feb Lecture 14: Pairs of random variables (Trinomial pmf)

- 10 Feb Tutorial-2

- 11 Feb Lecture 15: Joint CDF, Conditional PMFs, Independent random variables

- 15 Feb Lecture 16: Independent random variables

- 16 Feb Lecture 17: Independent random variables

- 17 Feb Tutorial-3

- 18 Feb Tutorial-2,3 (Discussion on Gambler's rain)

- 22 Feb Lecture 18: Continuous probability spaces

- 23 Feb No class

- 24 Feb Lecture 19: Probability measures on R

- 25 Feb Lecture 20: PDF and CDF

- 29 Feb Lecture 21: Continuous random variables

- 01 Mar Lecture 22: Pairs of random variables

- 02 Mar Lecture 23: Marginal pdfs, CDFs, conditional pdfs,CDFs

- 03 Mar Lecture 24: Conditional pdfs, CDFs for single and pair of random variables

- 07 Mar Lecture 25: Independent random variables

- 08 Mar Lecture 26: Mixed random variables, digital communication example

- 09 Mar Lecture 27: Mixed random variables, Functions of single random variable

- 10 Mar Tutorial-4

- 14 Mar Lecture 28: Functions of single random variables

- 15 Mar No class

- 16 Mar Lecture 29: Single function of two random variables

- 17 Mar Lecture 30: Functions of two random variables

- 21 Mar Lecture 31: Functions of two random variables

- 22 Mar Lecture 32: Expectation

- 23 Mar Lecture 33: Expectation

- 28 Mar Tutorial

- 30 Mar Lecture 34: Expectations with two random variables

- 31 Mar Tutorial

- 04 Apr Lecture 35: Expectations with two random variables

- 05 Apr Lecture 36: Conditional expectation

- 06 Apr Lecture 37: Conditional variance

- 07 Apr Lecture 37: Wrap up on conditional variance

- 11 Apr Lecture 38: Gaussian random variables

- 12 Apr Lecture 39: Characteristic functions

- 13 Apr Lecture 40: Characteristic functions

- 18 Apr Lecture 41: Asymptotic results (Weak law of large numbers)

- 19 Apr Lecture 42: Central limit theorem

- 21 Apr Tutorial