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

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