DEPARTMENT OF MATHEMATICS

MTL 106 (Introduction to Probability Theory and Stochastic Processes) 4 Credits (3-1-0)

I Semester 2024-2025

Lecture Classes (Slot D): Tuesday, Wednesday and Friday between 9:00 AM and 9:50 AM in LH 108.

Tutorial Classes: Mon, Tue, Thu and Fri between 2:00 PM and 2:50 PM in LH 313.1. Tutorial class will start from July 23rd 2024 (Tuesday).

For the students who registered II Semester 2023-2024, Re Major (for both I grade and E Grade) is scheduled on July 24th, 2024 (Wednesday) between 3 PM and 5 PM in my office.

Course Contents

Probability Theory: Axioms of probability, Probability space, Conditional probability, Independence, Baye's rule, Random variable, Some common discrete and continuous distributions, Distribution of Functions of Random Variable, Moments, Generating functions, Two and higher dimensional distributions, Functions of random variables, Order statistics, Conditional distributions, Covariance, correlation coefficient, conditional expectation, Modes of convergences, Law of large numbers, Central limit theorem.

(No. of Lectures - 28)

Stochastic Processes: Definition of Stochastic process, Classification and properties of stochastic processes, Simple stochastic processes, Stationary processes, Discrete and continuous time Markov chains, Classification of states, Limiting distribution, Birth and death process, Poisson process, Steady state and transient distributions, Simple Markovian queuing models (M/M/1, M/M/1/N, M/M/c/N, M/M/N/N).

(No. of Lectures - 14)

Main Text Books

1.      Introduction to Probability and Stochastic Processes with Applications, Liliana Blanco Castaneda, Viswanathan Arunachalam, Selvamuthu Dharmaraja, Wiley, Asian Edition, Jan. 2016.

2.      Probability and Statistics with Reliability, Queueing and Computer Science Applications, Kishor S. Trivedi, John Wiley, second edition, 2001.

3.      Introduction to Probability Models, Sheldon M. Ross, Academic Press, tenth edition, 2009.

Reference Books

1.      Introduction to Probability Theory and Stochastic Processes, Video course, NPTEL Phase II.

1.      Introduction to Probability Theory and Stochastic Processes (Tamil), Video course, NPTEL Phase II.

2.      Introduction to Probability, Statistical Methods, Design of Experiments and Statistical Quality Control, Dharmaraja Selvamuthu and Dipayan Das, Springer, second edition, 2024.

2.      An Introduction to Probability and Statistics, Vijay K. Rohatgi and A.K. Md. Ehsanes Saleh, John Wiley, second edition, 2001.

3.      Stochastic Processes, J. Medhi, New Age International Publishers, 3rd edition, 2009.

4.      Stochastic Processes, Video course, NPTEL Phase II.

5.      Probability, Random Variables and Stochastic Processes, Athanasios Papoulis and S. Unnikrishna Pillai, Tata Mcgraw-Hill, fourth edition, 2002.

6.      An Introduction to Probability Theory and its Applications, Vol. I & II, William Feller, Wiley Eastern, third edition, 2000.

Lecture Notes
Introduction to this course

Sl. No. Topics Videos Relevant Tutorial Sheet

1 Axioms of probability, Probability space, Conditional probability, Independence, Baye's rule Part 1, Part 2, Part 3, Part 4, Part 5 1

2 Random variable, Some common discrete and continuous distributions Part 1, Part 2, Part 3, Part 4, Part 5 Part 6, Part 7, Part 8, Part 9 Part 10, Part 11 2

3 Distribution of Functions of Random Variable, Moments, Generating functions Part 1, Part 2, Part 3, Part 4, Part 5 3

4 Two and higher dimensional distributions Part 1, Part 2, Part 3, Part 4, Part 5 4

5 Functions of random variables, Order statistics, Conditional distributions, Covariance, correlation coefficient, conditional expectation Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14 5

6 Modes of convergences, Law of large numbers, Central limit theorem Part 1, Part 2, Part 3, Part 4, Part 5 6

7 Definition of Stochastic process, Classification and properties of stochastic processes, Simple stochastic processes, Stationary processes Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13 7

8 Discrete time Markov chains, Classification of states, Limiting distribution Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12 8

9 Continuous time Markov chains, Limiting distribution, Birth and death process, Poisson process, Steady state and transient distributions Part 1, Part 2, Part 3, Part 4, Part 5, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10 9

10 Simple Markovian queuing models (M/M/1, M/M/1/N, M/M/c/N, M/M/N/N Part 1, Part 2, Part 3, Part 4, Part 5, Part 6 10

Note: The above classification of lecture notes are tentative only. (courtesy: NPTEL course)

Name	Room No.	Phone No.	Email
S Dharmaraja	MZ 164	7104	dharmar@maths.iitd.ac.in

Sl. No.	Topics	Videos	Relevant Tutorial Sheet
1	Axioms of probability, Probability space, Conditional probability, Independence, Baye's rule	Part 1, Part 2, Part 3, Part 4, Part 5	1
2	Random variable, Some common discrete and continuous distributions	Part 1, Part 2, Part 3, Part 4, Part 5 Part 6, Part 7, Part 8, Part 9 Part 10, Part 11	2
3	Distribution of Functions of Random Variable, Moments, Generating functions	Part 1, Part 2, Part 3, Part 4, Part 5	3
4	Two and higher dimensional distributions	Part 1, Part 2, Part 3, Part 4, Part 5	4
5	Functions of random variables, Order statistics, Conditional distributions, Covariance, correlation coefficient, conditional expectation	Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14	5
6	Modes of convergences, Law of large numbers, Central limit theorem	Part 1, Part 2, Part 3, Part 4, Part 5	6
7	Definition of Stochastic process, Classification and properties of stochastic processes, Simple stochastic processes, Stationary processes	Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13	7
8	Discrete time Markov chains, Classification of states, Limiting distribution	Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12	8
9	Continuous time Markov chains, Limiting distribution, Birth and death process, Poisson process, Steady state and transient distributions	Part 1, Part 2, Part 3, Part 4, Part 5, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10	9
10	Simple Markovian queuing models (M/M/1, M/M/1/N, M/M/c/N, M/M/N/N	Part 1, Part 2, Part 3, Part 4, Part 5, Part 6	10