# 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 Friday, November 29, 9:30-12:30am

## 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 Functions of random variables 5 26/8 2.5 Joint random variables Chap. 2: 49, 50, 53, 55 28/8 2.5 Independence Chap. 2: 56, 58, 59, 66 29/8 - Quiz 3 - 30/8 2.5 Covariance Chap 2: 54, 60, 62, 63 6 2/9 - Holiday 4/9 2.5 Change of variables formulas Chap 2: 68 5/9 - Quiz 4 - 6/9 2.6 Moment generating functions Chap 2: 67, 69, 70 7 9/9 2.8 Limit theorems Chap 2: 76, 78, 81, 83, 86 11/9 2.9 Stochastic processes none 12/9 - Quiz 5 - 13/9 3.1-3.2 Conditional probability Chap. 3: 1, 3, 5, 8 8 16/9 Class cancelled 18/9 3.3-3.4 Conditional expectation Chap. 3: 11, 12, 15, 19, 21, 26, 30 19/9 - Quiz 6 - 20/9 3.4 Conditional Variance formula Chap. 3: 36, 38, 40, 49, 50 9 23/9 No class (midterm week) 25/9 Midsemester exam, 9-11am, LH4 26/9 - No class (midterm week) - 27/9 No class (midterm week) 10 30/9 3.5 Probabilities by conditioning Chap. 3: 53, 54, 56, 60 2/10 - Holiday 3/10 - Quiz 7 - 4/10 4.1 Introduction to Markov chains Chap. 4: 1, 3, 4, 6 11 7/10 4.2 Chapman-Kolmogorov equation Chap. 4: 9, 12, 13 9/10 4.3 Communication classes Chap. 4: 15, 18(b), 21(a) 10/10 - Quiz 8 - 11/10 4.3 Recurrence and transience Chap. 4: 14, 16, 12 14/10 4.4 Long run proportions Chap. 4: 20, 21(b), 22, 24 16/10 4.4 Stationary distribution Chap. 4: 25, 26, 27, 28 17/10 - Quiz 9 - 18/10 4.6 Aperiodicity Chap. 4: 41, 42, 46, 47, 54, 59 13 21/10 4.7 Branching processes Chap. 4: 64, 66 23/10 4.8 Reversible Markov chains Chap. 4: 68, 70, 71, 73 24/10 - Quiz 10 - 25/10 5.1-5.2 Exponential distribution Chap. 5: 1, 3, 8, 10, 15 14 28/10 5.3 Poisson process Chap. 5: 32, 34, 37, 40 30/10 5.3 Interarrival and waiting times Chap. 5: 42, 44, 45, 57 31/10 - Quiz 11 - 1/11 - Holiday 15 4/11 6.1-6.2 Continuous-time Markov chains Chap 5: 59, 60, 64, 67 Chap 6: 1, 3, 4, 5 6/11 6.3 Birth-and-death processes Chap 6: 8, 9, 22, 25 7/11 - Quiz 12 - 8/11 6.4 Kolmogorov's equations Chap 6: 10, 11 16 11/11 6.5 Limiting probabilities Chap 6: 13, 16, 17, 19, 24 13/11 6.6 Time reversibility Chap 6: 29, 30, 32 14/11 - Quiz 13 - 15/11 Class cancelled 17 18/11 6.9 Matrix formulation None! 18 29/11 - Final Exam 9:30am-12:30pm Venue: LH 4