# Probability Theory - Math/Stats 525, Winter 2009

 Instructor: Roman Vershynin Office: 4844 East Hall E-mail: romanv "at" umich "dot" edu, to be used as an emergency contact only (e.g. sick during an exam). Any mathematical and administrative matters should be discussed in person during my office hours. Class meets: Tu, Th 1:10-2:30 in 1068 East Hall. Office Hours: M 3-4:30, W 1:30-3 in 4844 East Hall. Course description: This course is a thorough study of the mathematical theory of probability. This is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. We will cover: probability spaces, random variables, moment generating functions, limit theorems, conditional expectation, Markov chains, and Poisson processes.

Prerequisites and related courses: MATH 451 or 450. Also, MATH 425/STATS 425 would be very helpful. There is substantial overlap of our course with Math 425 (Intro. to Probability), but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Natural sequels are: Math 526 (Discr. State Stoch. Proc.), Stat 426 (Intro. to Math Stat.), and the sequence Stat 510 (Mathematical Statistics I)-- Stat 511 (Mathematical Statistics II).

Textbook: Sheldon M. Ross, Introduction to Probability Models, 9th edition, Elsevier, 2007.

• Quizzes (30%), every Thursday.
• Midterm Exam I (35%), Thursday, February 19, in class.
• Midterm Exam II (35%), April 21, in class.
There will be no final exam.

The guaranteed minimum grade will be determined by the following curve: A: 90 and up, B: 80 and up, C: 70 and up, D; 60 and up. You will be able to view your grades via Ctools. Auditors taking this course for credit will get a Pass for the grade C- or above.

Quizzes: The quizzes will take place at the beginning of the class on most Thursdays, for approximately 10 minutes. Each quiz will be based on the homework assigned for the previous week. There will be no make-up for the quizzes for any reason. One quiz with the lowest grade will be dropped.

Exams: Both exams are closed notes and closed book. Calculators are not permitted. You are allowed to bring one 3x5" index card with hand-written notes (both sides). There will be no make-up exams. A missed exam counts as 0 points, with the following exception. If you miss an exam due to a documented medical or family emergency, the exam's weight will be added to the other exam. If you miss both exams for whatever reason, the course is considered failed.

Practice Exams: These previous year exams may be based on material and/or textbooks different from this course, and they may not represent or resemble the actual exams in this course: Sample Exam 1, Sample Exam 2.

Lecture Schedule and Homework
Homework will be assigned every class but not collected or graded. It is a good idea to read the textbook ahead of lectures. Here is a link to the lecture schedule of my past MATH 525 course.

1. Jan. 8, Th. Sections 1.2-1.3. Homework 1: 5-10, 15, 48.
2. Jan. 13, Tu. Sections 1.4-1.5. Homework 2: 17 (first question), 19, 20, 28, 29.
3. Jan. 15, Th. Quiz 1 (1.2-1.5). Solutions. Section 1.6. Homework 3: 30, 36, 37, 39, 40, 43, 45.
4. Jan. 20, Tu. Sections 2.1-2.2.1. Homework 4: 2-6, 8, 9.
5. Jan. 22, Th. Quiz 2 (1.6). Solutions. Sectons 2.2.2-2.3. Homework 5: 14, 15, 16, 27, 30, 32, 33, 35.
6. Jan. 27, Tu. Section 2.4. Homework 6: 40, 45, 46, 49, 53.
7. Jan. 29, Th. Quiz 3 (2.1-2.3). Solutions. Section 2.5.1. Homework 7: 55, 58, 71.
8. Feb. 3, Tu. Sections 2.5.2-2.5.3. Homework 8: 37, 49, 50, 52, 53, 56, 57.
9. Feb. 5, Th. Quiz 4 (2.4-2.5.1). Solutions. Section 2.6. Homework 9: 51, 61, 60, 65, 74.
11. Feb. 12, Th. Quiz 5 (2.5.2-2.6). Solutions. Sections 3.1-3.2. Homework 11: from Chapter 2: 76, from Chapter 3: 1, 3, 4.
12. Feb. 17, Tu. Review for Midterm Exam I.
13. Feb. 19, Th. Midterm Exam I (click here). (covers Chapters 1 and 2). Solutions.
14. March 3, Tu. Sections 3.3-3.4. Homework 12: 5-15, 21.
15. March 5, Th. No quiz. Sections 3.4.1-3.5. Homework 13: 19, 22, 23, 24, 25, 27.
16. March 10, Tu. Section 4.1. Homework 14: Chapter 3: 37, 49; Chapter 4: 1-4.
17. March 12, Th. Quiz 6 (3.4-3.5). Solutions. Section 4.2. Homework 15: 5-11.
18. March 17, Tu. Section 4.3. Homework 16: 13-16.
19. March 19, Th. Quiz 7 (4.1). Solutions. Sections 4.3 (contd.)-4.4. Homework 17: Examples 4.17-4.19 on pp.206-207; Problems 18,19.
20. March 24, Tu. Sections 4.5.1, 4.8. Homework 18: 21, 23, 24, 27, 29, 30, 32, 33, 41, 42.
21. March 26, Th. Quiz 8 (4.2-4.4). Solutions. Section 5.2 (excluding 5.2.4). Homework 19: work out Examples 5.3, 5.9, 5.10, solve problems 1-7.
22. March 31, Tu. Sections 5.3.1-5.3.3 (begin). Homework 20: 37, 39, 40, 41, 49.
23. April 2, Th. Quiz 9 (4.8, 5.2). Solutions. Section 5.3.3 (contd) - 5.3.4 (Example 5.17: The Coupon Collecting Problem). Homework 21: 42, 43, 44, 47.
24. April 7, Tu. Sections 5.3.4 (Example 5.15) - 5.3.5 (Theorem 5.2, Example 5.20). Homework 22: 48(a), 50(a), 52, 53(a).
25. April 9, Th. Quiz 10 (5.3.1-5.3.4). Solutions. Monte Carlo methods. Metropolis Algorithm (incl. Section 4.9).
26. April 14, Th. Metropolis Algorithm, Gibbs Sampler, Simulated Annealing (continuation).
27. April 16, Th. Review for Midterm Exam II.
28. April 21, Th. Midterm Exam II (click here). (covers Chapters 3,4,5, excluding Lectures 25, 26).