# 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 Office hours: by request Teaching Assistants John Bruer (ACM), Yuton Chen (ACM), Lauren Eaton (EE), Alex Gittens (ACM) Office hours: Fri, 3-4 pm; Mon, 7-9 pm. Room 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

• 22 Sep 2011: added TAs and office hours. Established a Piazza account for the class.
• 17 Jul 2011: web page creation

### Lecture Schedule

W Date Topic Reading Homework
##### 1
27 Sep
29 Sep
Events, probabilities and random variables
• $$\sigma$$ fields and probability spaces
• Conditional probability, independence, Bayes' formula
• The law of averages
• 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
Survey

##### 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)

Gubner, Chapter 2-3

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

Gubner, Chapters 4, 5

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)

Gubner, Chapters 4, 5

HW 4
Survey

##### 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)

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

In addition, all students in the class must sign in at office hours at least once in the first three weeks of the course, or sign up for Piazza and post at least one question or response.

### Collaboration Policy

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

• ACM/EE 116 Piazza page - an online collaboration site for the course has been established using Piazza. This site can be used to post questions and give responses (from students or instructors). Postings can be anonymous if desired.

No collaboration is allowed on the ﬁnal exam.