Problem Sets for Math 180c, Spring 2009

Unless otherwise specified, homework is due on Thursdays by 8:00 pm in Vladimir Pesic’s homework box on the 6^th floor of AP&M.

Reading for Weeks 1 & 2: Taylor and Karlin: p. 333-349 & 355-364

Optional reading: Lawler: p. 65-82 and Durrett: p. 158-176

Problem set 1: Due Thursday April 9

T&K: VI.1 (p. 341- 344) Ex. 1.1 (answer for third is wrong in back of book), 1.2, 1.5, 1.6 (a,b only), Pr. 1.3, 1.8

T&K :VI.2 (p. 353 – 355) Ex. 2.2, 2.4, Pr. 2.2, 2.3

Also:

1. Let X(t) be a Poisson process with parameter lambda. Find the probability that the 5^th sojourn time ends before t, but the second sojourn time begins after t.

(No calculation needed; just think what it means.)

Solutions

Reading for Week 3: Taylor and Karlin p. 366-372 & 379-392

Problem set 2 (modified): Due Thursday April 16

T&K: VI.3: (p. 364-365) Ex. 3.1, 3.3; Pr. 3.1, 3.3, 3.4

T&K: VI.4: (p. 376-379) Ex. 4.2; Pr. 4.2

(Note: Ex. 4.4 is postponed to Problem set 3.)

Solutions

Reading for Week 4: Taylor and Karlin p. 394-408

Problem set 3: Due Monday, April 27, 8:00 pm

T&K: VI.4: (p. 376-379) Ex. 4.4, Pr. 4.3, 4.4, 4.6

T&K: (p. 392-394): Pr. 5.2

Also:

1. Let X(t) be a Poisson process with parameter lambda. Find the probability that the 5^th sojourn time begins before t, but the second sojourn time ends after t.

Solutions

First midterm Friday, April 24

Reading for Week 5: Taylor and Karlin p. 419-424

Problem set 4: Due Thursday, April 30

T&K: (p. 406-408): Ex. 6.2, Pr. 6.2, 6.4

Solutions

Notes on Markov chains with finitely many states.

Reading for Week 6: Taylor and Karlin p. 426-441

Optional reading: Lawler: p. 131-144, and Durrett: Chapter 5

Problem Set 5: Due Thursday, May 7

T&K: (p. 424-426) Ex. 1.2, 1.3; Pr. 1.1, 1.3

T&K: (p. 431-432: Ex. 2.1

T&K: (p. 435-437): Ex. 3.1, 3.3; Pr. 3.2

Solutions

Reading for Week 7: Taylor and Karlin p. 441 – 455

Problem Set 6: Due Thursday, May 14

T&K: (p. 435-437): Pr. 3.4

T&K: (p. 445-447) Ex. 4.2, 4.3, 4.5; Pr. 4.1, 4.5

T&K: (p. 455-457) Ex. 5.1; Pr. 5.1, 5.4

Solutions

Reading for Week 9: Taylor and Karlin p. 473-481, 541-555

Problem Set 7: Due Thursday, May 26

T&K: (p. 487-491): Ex. 1.1, 1.4, 1.5, Pr. 1.1, 1.3, 1.7

T&K: (p. 555-557): Ex. 2.1, 2.3, Pr. 2.4.

Solutions

Reading for Week 10: Taylor and Karlin p. 558 – 565

Problem Set 8: Due Thursday, June 4

T&K: (p. 555-557): Pr. 2.2.

T&K: (p. 566-567): Ex. 3.1, 3.3, 3.4, 3.5, Pr. 3.1.

Solutions