E2 202 : Random Processes, Fall 2018

Lectures

Homeworks

Tutorials

  • 04 Aug 2018: Sample space, sigma algebra, example constructions of sigma algebras, “infinitely often” and “all but finitely many” events (Notes and exercises)

  • 11 Aug 2018: Probability measure, continuity of probability, independence (Notes and exercises)

  • 18 Aug 2018: Conditional probability, conditional independence (Notes and exercises)

  • 22 Aug 2018: 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 (Notes and exercises)

  • 01 Sep 2018: Functions of random variables: sum, min, Jensen's inequality, Cauchy-Schwartz inequality (Notes and exercises)

  • 08 Sep 2018: Sigma algebra generated by a random variable, Simulation of CDFs on a computer, properties of joint CDFs, overview of Riemann and Lebesgue integrals, independence of random variables - definition and equivalent forms for jointly discrete and jointly continuous random variables (Notes) (Exercises)

  • 15 Sep 2018: Almost sure events, Markov's inequality, some properties of nonnegative random variables with finite expectation, self-independence of random variables (Notes and exercises)

  • 20 Sep 2018: Problems (Notes and exercises)

  • 29 Sep 2018: Transformations of RV ( Notes, Reference: Chapter 12 of this book).

  • 06 Oct 2018: Conditional expectation, law of iterated expectations, geometric interpretation of conditional expectation as MMSE estimate, example problems (Notes and Exercises)

  • 07 Oct 2018: Moment generating functions, characteristic functions, jointly Gaussian random variables (or Gaussian random vectors) (Notes and Exercises)

  • 13 Oct 2018: Convergence of real sequences, modes of convergence of a sequence of random variables, Borel-Cantelli lemma, examples (Notes)

  • 03 Nov 2018: Stopping times and basics of DTMCs (Notes)

  • 08, 10 Nov 2018: Problems on hitting times, recurrence, invariant distributions (Notes)

Tests

  • 01 Sep 2018: Quiz 1 Solution

  • 15 Sep 2018: Quiz 2 Solution

  • 22 Sep 2018: Mid Term 1 Solution

  • 13 Oct 2018: Quiz 3 Solution

  • 03 Nov 2018: Quiz 4 Solution

  • 10 Nov 2018: Quiz 5 Solution

  • 17 Nov 2018: Mid Term 2

  • 24 Nov 2018: Quiz 6

  • 06 Dec 2018: Final (Hours: 2pm - 5pm, Venue: EC 1.06,1.07,1.08)

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, moment generating function, characteristic function

  • 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 Information

Slack

Students can signup for course slack using their iisc.ac.in email at Slack signup.
Add yourself to the public channel #rp-2018.

Instructors

Utpal Mukherji
Office: ECE 1.02
Hours: By appointment.

Parimal Parag
Office: ECE 2.17
Hours: By appointment.

Time and Location

Classroom: ECE 1.08, Main ECE Building.
Class Hours: Tue/Thu 02:00 pm - 03:30 pm.
Tutorial Hours: Sat 2:30 pm - 4:00 pm.

Teaching Assistants

Karthik P N
Email: periyapatna@iisc.ac.in
Office: MP 327
Hours: Saturdays (after tutorial session)

Prathamesh Mayekar
Email: prathamesh@iisc.ac.in
Office: SP 1.17
Hours: Message me on slack (@Prathamesh Mayekar) or email me to schedule a meeting.

Textbooks

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

Discrete 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.