Probability Theory - Math/Stat 525, Fall 2008

Instructor: Roman Vershynin Office: 4844 East Hall` E-mail: romanv "at" umich "dot" edu Class meets: Tu, Th 11:40-1:00 in 1068 East Hall. Office Hours: Tu 2:30-4:00, We 1:30-3:00 in 4844 East Hall. After classes end:  Tuesday, Dec. 16: 2:30-4:00, Wednesday, Dec. 17: 1:30-3:00 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 is a strongly recommended prerequisite; MATH 425/STATS 425 would be 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.

Grading

• Quizzes (30%), every Thursday.
• Midterm Exam (30%), October 16, in class.
• Final Exam (40%), December 18, 10:30 am - 12:30 pm, in 260 Denn.
  This exam is closed notes and closed book. Calculators are permitted. You are also allowed to bring one 3x5" index card with hand-written notes (both sides).
  Practice problems: here ou can find two practice exams. Also, here is a plenty of practice problems.

Curve: The grade curve will be straight, i.e. your letter grade will be determined by the number of points you have earned. The actual grade cutoffs will be determined after the final exam. The plus/minus adjustments will be made on my discretion.

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.

Missed exams: There will be no make-up exams. A missed exam counts as 0 points. If you miss the midterm exam due to a documented medical or family emergency, the midterm's weight will be added to the final exam.

Auditors: The auditors taking this course for credit will get a Pass for the grade C- or above.

Announcement and Grade Posting: Announcements (such as change of office hours) will occasionally be e-mailed. You can view your grades and the list of announcements at Ctools site https://ctools.umich.edu/portal/site/70bdc55e-3535-4612-913f-b6eb58e041ac

Tentative Lecture Schedule and Homework

1. Sep. 4, Th. Sections 1.2-1.3. Homework 1: 5-10, 15, 48.
2. Sep. 9, Tu. Sections 1.4-1.5. Homework 2: 17 (first question), 19, 20, 28, 29.
3. Sep. 11, Th. Quiz 1 (1.2-1.5). Section 1.6. Homework 3: 30, 36, 37, 39, 40, 43, 45. Solutions to Homework 1, 2, Quiz 1.
4. Sep. 16, Tu. Sections 2.1-2.2.1. Homework 4: 2-6, 8, 9.
5. Sep. 18, Th. Quiz 2 (1.6). Sectons 2.2.2-2.3. Homework 5: 14, 15, 16, 27, 30, 32, 33, 35. Solutions to Homework 3, Quiz 2.
6. Sep. 23, Tu. Section 2.4. Homework 6: Work out Example 2.27 (try first without looking at the solution), and problems 40, 45, 46, 53.
7. Sep. 25, Th. Quiz 3 (2.1-2.3). Section 2.5.1. Homework 7: 55, 58, 71. Solutions to Homework 4, 5, Quiz 3.
8. Sep. 30, Tu. Section 2.5.2-2.5.3. Homework 8: 37, 49, 50, 52, 53, 56, 57.
9. Oct. 2, Th. Quiz 4 (2.4-2.5.1). Section 2.6. Homework 9: Example 2.39 (try without looking at the solution), 61, 60, 65, 74. Solutions to Homework 6, 7, Quiz 4.
10. Oct. 7, Tu. Section 2.7. Homework 10 (click here).
11. Oct. 9, Th. Quiz 5 (2.5.2-2.6). Sections 3.1-3.2. Homework 11: from Chapter 2: 76, from Chapter 3: 1, 3, 4. Solutions to Homework 8, 9, Quiz 5.
12. Oct. 14, Tu. Review for the Midterm Exam. Sample Exam 1, Sample Exam 2. These previous year exams may be based on material and/or textbooks different from this course, and they do not represent or resemble the actual exams in this course. Solutions to Homework 10.
13. Oct. 16, Th. Midterm Exam (covers Chapter 2). Solutions.
14. Oct. 23, Th. No quiz. Sections 3.3.-3.4. Homework 12: 5-15, 21.
15. Oct. 28, Tu. Sections 3.4.1-3.5. Homework 13: 19, 22, 23, 24, 25, 27.
16. Oct. 30, Th. Quiz 6 (3.2). Section 3.6.2 (optional). Homework 14: 37, 40, 49. Solutions to Homework 11, 12, 13, 14, Quiz 6.
17. Nov. 4, Tu. Section 4.1. Homework 15: 1-4.
18. Nov. 6, Th. Quiz 7 (3.4-3.5). Section 4.2. Solutions to Quiz 7. Homework 16: 5-10.
19. Nov. 11, Tu. Section 4.3. Homework 17: 13-16.
20. Nov. 13, Th. Quiz 8 (4.1). Section 4.3 (contd) - 4.4 (begin). Solutions to Homework 15, 16, 17, Quiz 8. Homework 18: Work out Examples 4.17, 4.18; Problems 18, 19.
21. Nov. 18, Tu. Section 4.4. Homework 19: 21, 23, 24, 27, 29, 30, 32, 33.
22. Nov. 20, Th. Quiz 9 (4.2-4.3). Sections 4.5.1, 4.7. Solutions to Homework 18, 19, Quiz 9. Homework 20: 35, 36, 42, 61.
23. Nov. 25, Tu. Section 5.2 (excluding 5.2.4). Homework 21: work out Example 5.3, solve problems 1-7.
24. Dec. 2, Tu. Section 5.3.1-5.3.2. Homework 22: 37, 39, 40, 41.
25. Dec. 4, Th. Quiz 10 (5.2). Section 5.3.3-5.3.4. Solutions to Homework 20, 21, Quiz 10. Homework 23: 42, 44, 47.
26. Dec. 9, Tu. Review for the Final Exam. Solutions to Homework 22, 23.

Useful links

• C.M.Grinstead and J.L.Snell, Introduction to Probability
• J.Walrand, Probability and Random Processes
• Virtual Laboratories in Probability and Statistics

Course webpage: http://www.umich.edu/~romanv/teaching/2008-09/525/525.html