# ACM/EE 116, Fall 2011

 Introduction to Probability and Random Processes with Applications Instructors Richard Murray (CDS/BE) Lectures: Tu/Th, 9-10:30, 105 ANB Teaching Assistants TBD Office hours: TBD

### Course Description

Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Random variables, expectation and conditional expectation, joint distributions, covariance, moment generating function, central limit theorem, weak and strong laws of large numbers, discrete time stochastic processes, stationarity, power spectral densities and the Wiener-Khinchine theorem, Gaussian processes, Poisson processes, Brownian motion. The course develops applications in selected areas such as signal processing (Wiener filter), information theory, genetics, queuing and waiting line theory, and finance.

### Announcements

• 17 Jul 2011: web page creation

### Lecture Schedule

 Date Topic Reading Homework 27 Sep 29 Sep Events, probabilities and random variables $$\sigma$$ fields and probability spaces Conditional probability and independence The law of large numbers Random variables (discrete and continuous) G&S, Chapters 1 and 2, Appendices Read Appendices A & B (history; 4 pages) Sections 1.1-1.5 (14 pages) Sections 2.1-2.3 (10 pages) HW 1 4 Oct 6 Oct Discrete random variables Probability mass functions Independence Expectation and moments Conditional distributions and conditional expectation Sums of random variables G&S, Chapter 3 Sections 3.1-3.8 (26 pages) HW 2 11 Oct 13 Oct Continuous random variables Probability density functions Independence Expectation and moments Conditional distributions and conditional expectation Functions of random variables Multivariate normal distribution G&S, Chapter 4 Sections 4.1-4.9 (30 pages) HW 3 18 Oct 20 Oct Generating functions and their applications Generating functions Random walks, branching processes Characteristic functions Law of large numbers, central limit theorem G&S, Chapter 5 Sections 5.1-5.4, 5.6A, 5.7-5.10 (48 pages) HW 4 25 Oct 27 Oct Markov processes Markov property, Markov chains Classification of states and chains Stationary distributions and the limit theorem Examples G&S Chapter 6 Sections 6.1-6.5 (37 pages) HW 5 1 Nov 3 Nov Convergence of random variables Modes of convergence Laws of large numbers The strong law Martingales G&S Chapter 7 Sections 7.1-7.5 (27 pages) HW 6 8 Nov 10 Nov* Random and stationary processes Stationary processes Examples: renewal processes, queues Wiener process Linear prediction Autocovariances and spectra Stochastic integration and the spectral representation Gaussian processes G&S Chapter 8, 9 Sections 8.1-8.5 (11 pages) Sections 9.1-9.4, 9.6 (28 pages) HW 7 15 Nov* 17 Nov Special Topics (instructor dependent) HW 8 22 Nov 29 Nov Special Topics (instructor dependent) HW 9 1 Dec Course review Final

### Textbook

The primary text for the course (available via the online bookstore) is

 [G&S] G. R. Grimmett and D. R. Stirzaker, Probability and Random processes, third edition. Oxford University Press, 2001.

The ﬁnal grade will be based on homework and a ﬁnal exam:

• Homework (75%) - There will be 9 one-week problem sets, due in class one week after they are assigned. Students are allowed three grace periods of two days each that can be used at any time (but no more than 1 grace period per homework set). Late homework beyond the grace period will not be accepted without a note from the health center or the Dean.
• Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely 3 hours in one sitting)