# MA 261: Probability Models

 Instructor: Arvind Ayyer Office: X-15 (new wing) Office hours: TBD Phone number: (2293) 3215 Email: (First name) at iisc dot ac dot in Class Timings: Mondays, Wednesdays and Fridays, 10:00 — 11:00 Classroom: LH-4 (new wing, first floor) Textbook: Introduction to Probability Models (11th edition) by Sheldon M. Ross Academic Press, 2014 ISBN-13 - 978-9351072249 Supplementary Texts: (a) Probability and random processes by Geoffrey R. Grimmett and David R. Stirzaker Oxford University Press, 2001 ISBN-13 - 978-0198572220 (b) Markov Chains and Mixing Times by David A. Levin, Yuval Peres and Elizabeth L. Wilmer Markov Chains and Mixing Times ISBN-13 - 978-0812847398 TA: Dipankar Roy (dipankarroy at iisc dot ac dot in) Tutorials: Thursdays 9:30 — 10:00

## Course Description

Sample spaces, events, probability, discrete and continuous random variables, Conditioning and independence,
Bayes' formula, moments and moment generating function, characteristic function, laws of large numbers,
central limit theorem, Markov chains, Poisson processes.

## Prerequisites

Basic linear algebra and some exposure to proofs and abstract mathematics.

## Exams

All exams will be closed book, closed notes, and
no calculators or electronic devices are allowed (no cell/smart phones).
No communication among the students will be tolerated.
There will be no make up exams.

The date for the midterms and final will be announced later.

Here are the weights for the homework and exams.
All marks will be posted online on Moodle.

• 5% – Attendance
• 15% – Quizzes
• 30% – Midterms
• 50% – Final

## Tentative Class Plan

#### Tutorials are marked in green.

 week date sections material covered homework and other notes 1 2/8 1.1-1.2 Basic set theory Chap. 1: 1, 3, 4, 5, 6 2 5/8 1.3-1.4 Probabilities Chap. 1: 8, 11, 12, 13, 15, 19, 21 7/8 1.5-1.6 Independence Chap. 1: 36, 37, 40, 43, 45, 47 8/8 - Quiz 1 - 9/8 - Holiday 3 12/8 - Holiday 14/8 2.1-2.2 Discrete random variables Chap. 2: 1, 2, 4, 5, 9, 16, 17, 20, 30 15/8 - Holiday - 16/8 2.3 Continuous random variables Chap. 2: 33, 34, 35, 36, 38 4 19/8 2.4 Expectation Chap. 2: 39, 40, 41, 47 21/8 2.5 Functions of random variables Chap. 2: 46, 47, 48 22/8 - Quiz 2 - 23/8 2.5 Joint random variables Chap. 2: 49, 50, 53, 55 5 26/8 2.5 Independence and covariance 28/8 2.6 Moment generating functions 29/8 - Quiz 3 - 30/8 2.6 Moment generating functions 6 2/9 - Holiday 4/9 2.8 Limit theorems 5/9 - Quiz 4 - 6/9 2.9 Stochastic processes 7 9/9 3.1-3.3 Conditional probability 11/9 3.4 Expectations by conditioning 12/9 - Quiz 5 - 13/9 3.4 Conditional Variance formula 8 16/9 3.5 Probabilities by conditioning 18/9 3.6 A random graph model 19/9 - Quiz 6 - 20/9 4.1-4.2 Introduction to Markov chains 9 23/9 No class (midterm week) 25/9 Midsemester exam 26/9 - No class (midterm week) - 27/9 No class (midterm week) 10 30/9 4.2 Restrictions of Markov chains 2/10 - Holiday 3/10 - Quiz 7 - 4/10 4.3 Classification of states 11 7/10 4.4 Long run proportions 9/10 4.4 Stationary distribution 10/10 - Quiz 8 - 11/10 4.6 Examples of Markov chains 12 14/10 16/10 17/10 - Quiz 9 - 18/10 13 21/10 23/10 24/10 - Quiz 10 - 25/10 14 28/10 30/10 31/10 - Quiz 11 - 1/11 - Holiday 15 4/11 6/11 7/11 - Quiz 12 - 8/11 16 11/11 13/11 14/11 - Quiz 13 - 15/11 16 18/11 20/11 21/11 - Quiz 14 - 22/11