MA 261: Probability Models

Instructor: Arvind Ayyer
Office: X-15 (new wing)
Office hours: By appointment
Phone number: (2293) 3215
Email: (First name) at iisc dot ac dot in
Class Timings: Mondays, Wednesdays and Fridays, 10:00 — 11:00
Classroom: Online (Microsoft Teams)
To attend the course,

IISc students: See the details here

Non-IISc students: Send me an email
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:
  • Ajay Kumar Nair (ajaynair at iisc dot ac dot in)
Tutorials: Thursdays 12:30 — 1:00pm for the quizzes

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.

Grading

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

Tentative Class Plan

Tutorials are marked in green.

week date sections material covered homework and other notes
1 2/10 -

Holiday
2 5/10 1.1-1.2 Basic set theory Chap. 1: 1, 3, 4, 5, 6, 8
7/10 1.3-1.4 Probabilities Chap. 1: 11, 12, 13, 15, 19
8/10 -

No quiz

 -
9/10 1.5-1.6 Conditional probability and independence Chap. 1: 21, 25, 28, 30, 31, 32
3 12/10 2.1 Random variables Chap. 1: 36, 37, 40, 43, 45, 47
Chap. 2: 1, 2, 4, 5, 6
14/10 2.2-2.3 Discrete random variables Chap. 2: 9, 16, 17, 20, 23
15/10 -

Quiz 1

 -
16/10 2.4 Continuous random variables Chap. 2: 27, 30, 33, 34, 35, 36, 38
4 19/10 2.5 Expectation of random variables Chap. 2: 39, 40, 46(a), 47
21/10 2.5 Functions of random variables Find expectations of the r.v.'s in Chap. 2: 33, 34, 35, 38
22/10 -

Quiz 2

 -
23/10 2.5 Joint random variables Chap. 2: 49, 50, 53, 55
5 26/10 -

Holiday
28/10 2.5 Joint random variables Grimmett and Stirzaker: Section 3.6, Problems 2,3
Grimmett and Stirzaker: Section 4.5, Problems 2,3
29/10 -

Quiz 3

 -
30/10 -

Holiday
6 2/11 2.5 Independence and Covariance Chap 2: 54, 56, 60, 62
4/11 2.5 Correlation coefficient Grimmett and Stirzaker: Section 3.6, Problem 4
Grimmett and Stirzaker: Section 4.5, Problem 4
5/11 -

Quiz 4

 -
6/11 2.6 Change of variables formulas Chap 2: 68
Grimmett and Stirzaker: Section 4.8, Problems 1, 2, 3, 8
7 9/11 2.8 Moment generating functions Chap 2: 67, 69, 70, 71
11/11 2.8 Limit theorems Chap 2: 77, 78, 81, 83, 86
12/11 -

Quiz 5

 -
13/11 2.9 Stochastic processes Chap 2: Example 2.53
8 16/11 3.1-3.2 Conditional probability Chap. 3: 1, 3, 5, 8
18/11 3.3-3.4 Conditional expectation Chap. 3: 11, 12, 15, 19, 21, 26, 30
19/11 -

Quiz 6

 -
20/11 3.4 Conditional Variance formula Chap. 3: 36, 37, 38, 40, 44
9 23/11

Midsemester exam, 9:00-11:00am
25/11

No class (midterm week)
26/11 -

No class (midterm week)

 -
27/11

No class (midterm week)
10 30/11 3.5 Probabilities by conditioning Chap. 3: 50, 53, 56, 57, 60
2/12 4.1 Introduction to Markov chains Chap. 4: 1, 3, 4, 6
3/12 -

Quiz 7

 -
4/12 4.2 Chapman-Kolmogorov equation Chap. 4: 9, 12, 13
11 7/12 4.3 Communication classes Chap. 4: 15, 18(b), 21(a)
9/12 4.3 Recurrence and transience Chap. 4: 14, 16, 17
10/12 -

Quiz 8

 -
11/12 4.4 Long run proportions Chap. 4: 20, 21(b), 22, 24
12 14/12 4.4 Stationary distribution Chap. 4: 25, 26, 27, 28, 30
16/12 4.6 Aperiodicity Chap. 4: 41, 42, 46, 47, 54, 59
17/12 -

Quiz 9

 -
18/12 4.7 Branching processes Chap. 4: 64, 65, 66
13 21/12 4.8 Reversible Markov chains Chap. 4: 68, 70, 71, 75
23/12 5.1-5.2 Exponential distribution Chap. 5: 1, 3, 8, 10, 15
24/12 -

Quiz 10

 -
25/12 -

Holiday
14 28/12 5.3 Poisson process Chap. 5: 32, 34, 37, 40, 41
30/12 5.3 Interarrival and waiting times Chap. 5: 43, 44, 45, 57, 59, 60, 64, 67
31/12 -

Quiz 11

 -
1/1 -

Holiday
15 4/1 6.1-6.3 Continuous-time Markov chains Chap 6: 1, 3, 4, 5, 6
6/1 6.4 Kolmogorov's equations Chap 6: 8, 9, 10, 11
7/1 -

Quiz 12

 -
8/1 6.5 Limiting probabilities Chap 6: 13, 16, 17, 19, 22, 24, 25
16 11/1 6.6 Time reversibility Chap 6: 29, 30, 32
13/1 6.9 Matrix formulation Chap 6: 35, 38, 40
14/1 -

Quiz 13

 -
15/1 - Miscellaneous None!
17 22/1 -

Final Exam 9:30am-12:30pm