Math/Stats 525: Probability Theory — Fall 2019


Instructor: Michael Zieve   
Remaining Office Hours: Thursday 11:30am–1pm, Sunday 6pm–9pm, Monday 5pm–10pm in 3835 East Hall
Class: Tuesday and Thursday, section 1: 8:30–10:00 in 2306 Mason Hall; section 2: 10:00–11:30 in 1427 Mason Hall

For the course logistics including details about grading and expected work, see the syllabus.

Lecture Schedule and Homework Assignments:
  1. Sep. 3, Tu.  Sections 1.2–1.5.  Homework: 6, 7, 8, 10, 13, 15, 39 of chapter 1
  2. Sep. 5, Th.  Sections 1.6–2.2.  Homework: 18, 20 of chapter 1, and 2, 3, 8, 9 of chapter 2
  3. Sep. 10, Tu.  Sections 2.3–2.4.1.  Homework: 14, 16, 30, 32, 33 of chapter 2
  4. Sep. 12, Th.  Sections 2.4.2–2.5.2.  Homework: 40, 49, 53, 56, 58 of chapter 2
  5. Sep. 17, Tu.  Section 2.5.3.  Homework: 42, 51, 55, 59 of chapter 2 (and the variance in 56 and 58 if it wasn't done correctly last week)
  6. Sep. 19, Th.  Sections 2.5.4–2.6.  Homework: 67, 69 of chapter 2
  7. Sep. 24, Tu.  Section 2.6.  Homework: 63, 70, 74 of chapter 2
  8. Sep. 26, Th.  Section 2.6.  Homework: 81, 82 of chapter 2
  9. Oct. 1, Tu.  Section 2.8.  Homework: 77, 78(a) of chapter 2
  10. Oct. 3, Th.  Section 2.8.  Homework: 78(b), 79, 86 of chapter 2
  11. Oct. 8, Tu.  Sections 3.1–3.2. Homework: prepare for the midterm
  12. Oct. 10, Th.  Section 3.3. Homework: prepare for the midterm
  13. Oct. 15, Tu.  No class: Fall Study Break
  14. Oct. 17, Th.  Midterm, covering chapters 1 and 2
  15. Oct. 22, Tu.  Section 3.. Homework: 3, 4, 7, 11 of chapter 3
  16. Oct. 24, Th.  Sections 3.4–3.5. Homework: 15, 26, 27 of chapter 3
  17. Oct. 29, Tu.  Sections 4.1–4.2.  Homework: 5, 6, 8, 14, 16 of chapter 4
  18. Oct. 31, Th.  Section 4.3.  Homework: 12, 15, 18 of chapter 4
  19. Nov. 5, Tu.  Section 4.4.  Homework: 28, 45 of chapter 4
  20. Nov. 7, Th.  Section 4.4.  Homework: 24, 31, 33, 36 of chapter 4
  21. Nov. 12, Tu.  Section 4.8.  Homework: 49, 52 of chapter 4; also determine all stationary distributions of all irreducible Markov chains having a finite number of states and satisfying Σi Pij=1 for each j [Note: to do this problem, it is not enough to consider one such Markov chain, instead you must consider every such Markov chain]
  22. Nov. 14, Th.  Section 4.8.  Homework: 70, 71, 74 of chapter 4
    Extra credit (can be handed in on Tu. Dec. 3; please do not look for the answer online or ask anyone about this problem): consider a Markov chain on the nonnegative integers such that Pij=0 when |i-j|>1 and Pij>0 when |i-j|=1 (for i,j≥0). Give necessary and sufficient conditions (depending only on the values Pi,i+1 and Pi,i-1) for this chain to be each of the following: transient, recurrent, positive recurrent, null recurrent. Give an explicit example of a null recurrent Markov chain.
  23. Nov. 19, Tu.  Section 4.9.  Homework: Describe (with proof) exactly when the Markov chain defined on pp. 247–248 is irreducible, and when it has period 1 (i.e., is aperiodic).   (A remark: if both of these conditions hold then, since the Markov chain has a stationary distribution, the Markov chain converges to its stationary distribution for any choice of probability distribution on X0.)
  24. Nov. 21, Th.  Section 5.2.2.  Homework: 1, 2, 3, 4 of chapter 5
  25. Nov. 26, Tu.  Section 5.2.3.  Suggested practice problems (not to be handed in, but this is material you should know for the midterm): 9, 30 of chapter 5
  26. Nov. 28, Th.  No class: Thanksgiving
  27. Dec. 3, Tu.  Review.
  28. Dec. 5, Th.  Review for midterm
  29. Dec. 10, Tu.   Review for midterm
  30. Dec. 10, Tu.   6pm–8pm   Midterm in CHEM 1210, covering chapters 3–5, in addition to the Central Limit Theorem.