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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