# 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), Yutong Chen (ACM), May Eaton (EE), Alex Gittens (ACM) Office hours: Fri, 3-4 pm, 213 ANB; Mon, 7-9 pm, 107 ANB.

### 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

• 5 Dec 2011: Partial solutions for HW #9 are posted; will try to post additional solutions by Tue (6 Dec) evening.
• 2 Dec 2011: The final exam is available for pickup outside 107 Steele. (Make sure to get the right exam! The ACM/EE 116 and CDS 110 exams look similar.)
• 1 Dec 2011: HW #8 is graded. Scores: mean= 46/50, std dev = 6.1. Hours: mean=8.5, std dev = 2.9 (median=8)
• 27 Nov 2011: Homework deadlines and office hours for this week:
• Office hours: Tue (29 Nov), 3-4 pm and Thu (1 Dec) 7-9 pm on first floor Annenberg
• Homework deadlines: HW #8 extensions are due by 29 Nov, 10:30 am. HW #9 is due 2 Dec (Fri) @ 5 pm (auto-extension)
• Turn in homework in class or to the box outside 109 Steele
• 21 Nov 2011: HW #7 is graded. Scores: mean=45/50, std dev = 6.7. Hours: mean = 11.8, std dev = 5.5 (median = 10)
• 14 Nov 2011: HW #6 is graded. Scores: mean=43/50, std dev = 6.2. Hours: mean = 10.5, std dev = 4.5 (median = 9.5)
• 7 Nov 2011: HW #5 is graded (except extensions). Scores: mean = 42/50, std dev = 5.6. Hours: mean = 10.7, std dev = 4.3
• 31 Oct 2011: HW #4 is graded. Scores: mean = 46/50, std dev = 5.2. Hours: mean = 7.7, std dev = 3.5
• 24 Oct 2011: HW #3 is graded. Scores: mean = 44/50, std dev = 5.1. Hours: mean = 6.2, std dev = 3.3

### 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 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

##### 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

##### 3
11 Oct
13 Oct
Continuous random variables
• Probability density functions
• Independence
• Expectation and moments
• Conditional distributions and conditional expectation
• Functions of random variables
G&S, Chapter 4
• Sections 4.1-4.9 (30 pages)

Gubner, Chapters 4, 5

##### 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

##### 5
25 Oct
27 Oct
Convergence of random variables/processes
• Modes of convergence
• Borel-Cantelli lemmas
• Laws of large numbers
• The strong law
• Prediction and conditional expectation
• Monte Carlo simulation
G&S Chapter 7
• Sections 7.1-7.5, 7.9 (35 pages)

Gubner, Chapters 13, 14

##### 6
1 Nov
3 Nov
Introduction to random processes
• Discrete and continuous time processes
• Markov processes/chains (overview)
• Poison processes, birth-death processes
• Properties of random processes (mean, covariance, time correlation...)
G&S Chapters 8
• Sections 6.1, 6.8, 6.11
• Sections 8.1-8.2
##### 7
8 Nov
10 Nov*
Stationary processes and renewal processes
• Stationary processes
• Linear prediction
• Spectral density
• Renewal processes
G&S Chapter 9 and 10
• Sections 9.1-9.3, 9.5
• Sections 10.1-10.2, 10.4

HW 7
Soln 7 (partial)

##### 8
15 Nov*
17 Nov
Stochastic systems
• Multi-variate normal distributions
• Discrete-time Gaussian processes
• Continuous-time Gaussian processes
• Linear stochastic systems with Gaussian noise
• Random processes in the frequency domain
• Kalman filtering (if time)
G&S + Course notes

Gubner, Chapter 10

##### 9
22 Nov
29 Nov
Diffusion processes
• Brownian motion, Wiener process
• 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)

Gubner, Chapter 11

• Section 11.3

HW 9
Soln 9 (partial)

##### 10
1 Dec Course review

Review for final (watch out for typos!)

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 (or the mailbox outside 109 Steele) 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)

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 (eg, Course Hero, instructors manuals, other course web sites) 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.

## Old Announcements

• 17 Jul 2011: web page creation
• 22 Sep 2011: added TAs and office hours. Established a Piazza account for the class.
• 26 Sep 2011: Background survey and lecture 1 slides are posted
• 4 Oct 2011: HW #2 is now posted; due 11 Oct 2011 in class
• 5 Oct 2011: Office hour locations have moved (to rooms with more board space and seats)
• 7 Oct 2011: HW #1 is graded. Scores: mean = 64/70, std dev = 5.4. Hours: mean = 5.9, std dev = 3.0
• 12 Oct 2011: HW #3 is now posted
• 18 Oct 2011: HW #4 is now posted
• 18 Oct 2011: HW #2 is graded. Scores: mean = 63/70, std dev = 5.3. Hours: mean = 8.2, std dev = 4.3
• 25 Oct 2011: HW #5 is posted as well as the midterm course survey
• 1 Nov 2011: HW #6 is now posted
• 8 Nov 2011: HW #7 is posted.