E2 202, Fall 2017

Random Processes

Lectures

• 03 Aug 2017: Lecture-01 Preliminaries
• 08 Aug 2017: Lecture-02 Probability laws
• 10 Aug 2017: Lecture-03 Conditional Events
• 17 Aug 2017: Lecture-04 Random Variables
• 22 Aug 2017: Lecture-05 Random Vectors
• 24 Aug 2017: Lecture-06 Expectation
• 29 Aug 2017: Lecture-07 Generalized Expectations
• 31 Aug 2017: Lecture-08 Conditional Expectations
• 05 Sep 2017: Lecture-09 Joint Distributions
• 07 Sep 2017: Lecture-10 Transformations of Random Variables
• 12 Sep 2017: Lecture-11 Characteristic Function
• 14 Sep 2017: Lecture-12 Jointly Gaussian Random Variables
• 19 Sep 2017: Lecture-13 Convergence of Sequence of Random Variables - 1
• 26 Sep 2017: Lecture-14 Convergence of Sequence of Random Variables - 2
• 28 Sep 2017: Lecture-15 Convergence of Sequence of Random Variables - 3
• 05 Oct 2017: Lecture-16 Limit Theorems and the Laws of Large Numbers
• 07 Oct 2017: Lecture-17 Random Processes: Independence, Conditional Expectation, Filtration
• 10 Oct 2017: Lecture-18 Tractable Random Processes
• 12 Oct 2017: Lecture-19 Bernoulli Process
• 17 Oct 2017: Lecture-20 Stopping Time Sigma Algebras
• 19 Oct 2017: Lecture-21 Success Instants Process
• 24 Oct 2017: Lecture-22 Introduction to Markov Chains
• 26 Oct 2017: Lecture-23 Markov Chains
• 31 Oct 2017: Lecture-24 Markov Chains: Hitting and Recurrence Times
• 02 Nov 2017: Lecture-25 Markov Chains: Class Properties
• 03 Nov 2017: Lecture-26 Markov Chains: Invariant Distribution
• 14 Nov 2017: Lecture-27 Poisson Processes
• 16 Nov 2017: Lecture-28 Poisson Processes: Characterizations
• 17 Nov 2017: Lecture-29 Poisson Processes: Properties
• 21 Nov 2017: Lecture-30 Poisson Processes: Compound and Non-stationary
• 23 Nov 2017: Lecture-31 Poisson Processes: Points in Higher Dimensions

Homeworks

• 17 Aug 2017: Homework 1
• 31 Aug 2017: Homework 2
• 13 Sep 2017: Homework 3
• 27 Sep 2017: Homework 4
• 17 Oct 2017: Homework 5
• 31 Oct 2017: Homework 6
• 16 Nov 2017: Homework 7
• 27 Nov 2017: Homework 8

Tutorials

• 05 Aug 2017: Sample space, sigma algebra, example constructions of sigma algebras, “infinitely often” and “all but finitely many” events
• 12 Aug 2017: Probability measure, continuity of probability, independence and conditional independence - examples and exercises
• 19 Aug 2017: Borel sigma algebra, random variables, CDF and its properties, construction of CDF from a probability measure, example constructions of random variables with respect to various sigma algebras - exercises
• 24 Aug 2017: Random variables - examples and exercises
• 31 Aug 2017: Expectation of random variables, an overview of Lebesgue integrals
• 06 Sep 2017: Joint CDF, marginals, conditional distribution, calculating expectation of some commonly used random variables - exercises
• 09 Sep 2017: Simulation of random variables, finiteness of moments, expectation in terms of CDF for nonnegative random variables
• 16 Sep 2017: Transformation of random variables - exercises
• 23 Sep 2017: Moment generating function, characteristic function, Gaussian random vectors - exercises
• 28 Sep 2017: Convergence of real sequences, convergence of sequence of random variables, Borel-Cantelli Lemma
• 03 Oct 2017: Convergence of real sequences, convergence of sequence of random variables, Borel-Cantelli Lemma (contd.) - exercises
• 14 Oct 2017: Conditional expectations, an overview of stochastic processes, finite dimensional distributions and filtrations
• 19 Oct 2017: Stopping times, stopping time sigma algebra, strong Markov property
• 28 Oct 2017: Discussion - exercises
• 04 Nov 2017: Discrete Time Markov Chains and properties - examples
• 07 Nov 2017: Discrete Time Markov Chains: Invariant distributions
• 11 Nov 2017: Discussion of mid term 1 and mid term 2 solution
• 18 Nov 2017: Poisson processes - characterizations, interarrival times
• 25 Nov 2017: Poisson processes - merging, splitting, non-homogeneous and compound; Strong Markov property, Order statistics, summary of the course

Tests

• 31 Aug 2017: Quiz 1
• 09 Sep 2017: Quiz 2
• 21 Sep 2017: Mid Term 1
• 11 Oct 2017: Quiz 3
• 28 Oct 2017: Quiz 4
• 09 Nov 2017: Mid Term 2
• 22 Nov 2017: Quiz 5
• 07 Dec 2017: Final (Hours: 2pm - 5pm, Venue: EC 1.07, 1.08)

Grading Policy

Mid Term 1: 15\%
Mid Term 2: 15\%
Quizzes: 20\%
Final: 50\%

Course Syllabus

• Probability Theory:
  • axioms, continuity of probability, independence, conditional probability.
• Random variables:
  • distribution, transformation, expectation.
• Random vectors:
  • joint distribution, conditional distribution, expectation, Gaussian random vectors.
• Convergence of random sequences:
  • Borel-Cantelli Lemma, laws of large numbers, central limit theorem, Chernoff bound.
• Discrete time random processes:
  • ergodicity, strong ergodic theorem, definition, stationarity, correlation functions in linear systems, power spectral density.
• Structured random processes:
  • Bernoulli processes, independent increment processes, discrete time Markov chains, recurrence analysis, Foster’s theorem, reversible Markov chains, the Poisson process.

Course Description

Basic mathematical modeling is at the heart of engineering. In both electrical and computer engineering, uncertainty can be modeled by appropriate probabilistic objects. This foundational course will introduce students to basics of probability theory, random variables, and random sequences.

Slack/GitHub Information

Slack

Students can signup for slack using their iisc.ac.in email at Slack signup.

Instructors

Utpal Mukherji
Office: ECE 1.02

Parimal Parag
Office: ECE 2.17
Hours: Fri 02:00 pm - 03:00 pm.

Time and Location

Classroom: ECE 1.08, Main ECE Building.
Class Hours: Tue/Thu 02:00 pm - 03:30 pm.
Tutorial Hours: Sat 10:00 am - 11:30 am.

Teaching Assistants

Karthik P N: periyapatna@iisc.ac.in
Office: MP 327
Office Hours: Mon/Wed 03:45 pm - 04:45pm
Sahasranand KR: sahasranand@iisc.ac.in
Office: EC 2.19
Office Hours: Fridays, 1:30-3:30pm, EC 2.19

Textbooks

Probability and Random Processes, Geoffrey Grimmett and David Stirzaker, 3rd edition, 2001.

Event Stochastic Processes, Anurag Kumar, Department of Electrical Communication Engineering, Indian Institute of Science.

Random Processes for Engineers, Bruce Hajek, 2014.

Introduction to Probability, Dimitri P. Bertsekas and John N. Tsitsiklis, 2nd edition, 2008.

A First Course in Probability, Sheldon M. Ross, 2013.

Probability Essentials, Jean Jacod & Philip Protter, Springer, 2004.

Probability, Random processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance, Kobayashi, Hisashi, Brian L. Mark, and William Turin, Cambridge University Press, 2011.