# Difference between revisions of "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 John Bruer (ACM), Yuton Chen (ACM), Lauren Eaton (EE), Alex Gittens (ACM) 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

##### 1
27 Sep
29 Sep
Events, probabilities and random variables
• $$\sigma$$ fields and probability spaces
• Conditional probability, independence, Bayes' formula
• The law of large numbers
• Random variables (discrete and continuous)
G&S, Chapters 1 and 2, Appendices
• Optional: Read Appendices III and IV (history; 4 pages)
• Sections 1.1-1.5 (14 pages)
• Sections 2.1-2.3 (10 pages)

Gubner, Chapters 1 and 2 (partial)

• Sections 1.1-1.6 + Chapter 1 notes
• Section 2.1 (random variables)

HW 1

##### 2
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

##### 3
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)
• Supplemental notes

HW 3

##### 4
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

##### 5
25 Oct
27 Oct
Convergence of random variables/processes
• Modes of convergence
• Borel-Cantelli lemmas
• Laws of large numbers
• The strong law
• Monte Carlo simulation
G&S Chapter 7
• Sections 7.1-7.5 (27 pages)

HW 5

##### 6
1 Nov
3 Nov
Introduction to random processes
• Discrete and continuous time processes
• Markov processes/chains (overview)
• Poison processes
• Properties of random processes (mean, covariance, time correlation...)
• Examples and applications
G&S Chapters 8
• Sections 6.1 (Markov processes; 5 pages)
• Sections 8.1-8.6 (14 pages)
• Supplementary notes (OBC, Ch 4)

HW 6

##### 7
8 Nov
10 Nov*
Discrete time stochastic processes
• Stationary processes
• Examples: renewal processes, queues
• Linear prediction
G&S Chapter 9
• Sections 9.1-9.2, 9.5 (17 pages)
• Supplementary notes (OBC, Ch 4)

HW 7

##### 8
15 Nov*
17 Nov
Continuous time stochastic processes
• Wiener process
• Ornstein-Uhlenbeck process
• Stochastic integration and the spectral representation
• Linear stochastic systems
G&S Chapter 9
• Sections 9.3-9.4, 9.6 (17 pages)
• Supplementary notes (OBC, Ch 4)

HW 8

##### 9
22 Nov
29 Nov
Diffusion processes
• Brownian motion
• Diffusion properties, first passage times
• Stochastic calculus
• Ito integral, Ito formula (if time)
G&S Chapter 13
• Sections 13.1-13.4 (27 pages)
• Sections 13.7-13.9 (10 pages)

HW 9

##### 10
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 following additional texts may be useful for some students (on reserve in SFL):

 [Gubner] J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press, 2006. [S&W] H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing, third edition. Prentice Hall, 2002.

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)