Problem Sets for Math 180b, Winter 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 Week 1: Grinstead and Snell “Introduction to Probability” (Click here for free download) p. 133-150

Problem set 1. Due Thursday January 8.

p. 150 (Grinstead and Snell): 3(c,d), 14, 18, 22, 36.

Subsequent assignments will be longer!

Solutions

Reading for Week 2: Taylor and Karlin: 57-61, 70-73, 95-97, and Notes on Random Walks

Problem set 2 (corrected). Due Thursday, January 15.

From T&K: p. 61 Ex. 1.2, 1.5; p. 62: Prob. 1.3, 1.5; p. 77: Ex. 3.1; p. 98: Ex. 1.1

From online text by Durrett: p. 114: 60, 62, 64

Also (from Ross p. 414):

53: A prisoner is trapped in a cell containing 3 doors. The first door leads to a tunnel that returns him to his cell after 2 days travel. The second leads to a tunnel that returns him to his cell after 4 days travel. The third door leads to freedom after 1 day of travel. If it is assumed that the prisoner will always select doors with respective probabilities .5, .3, and .2, what is the expected number of days until the prisoner reaches freedom?

57: Suppose that the expected number of accidents per weeek at an industrial plant is 5. Suppose also that the numbers of workers injured in an accident each week are independent random variables with a common mean of 2.5. If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week.

Solutions

Reading for Week 3: T&K: p.100-102, Durrett: Chapter 4, Section 4.1 (once over lightly) and expanded Notes on Random Walks .

Problem set 3. Due Thursday, January 22.

Durrett: p. 141: 1,2

T&K p. 98: Ex 1.2, 1.3, 1.4; p. 99: Problems 1.2 & 1.4; p. 102: Ex. 2.1

Also:

1. Compute the variance of the time to freedom in Problem 53 from Problem set 2 above.

2. Suppose that Jack and Jill start with $7 and $8 respectively and play $1 bets (with p = q = 1/2) until one of them runs out of money. What is the probability that Jack will win all of Jill’s money? What is the expected number of bets until the game ends?

3. Same game, but Jack starts with $5, Jill with $10, and the probability p that Jack will win a bet is p=.6 (so he loses each bet with probability .4). Answer the two questions given in 2.

Here is a handy website for matrix calculations: http://www.quickmath.com/. This will be useful for Markov chain calculations.

Solutions

Reading for Week 4: T&K p 105-112 and 116- 127

Problem set 4. Due Thursday, January 29.

T&K:

p. 102-105: Ex. 2.2, Pr. 2.3, 2.5

p. 112-116: Ex. 3.1, 3.2, Pr. 3.1, 3.4, 3.7

p. 127-130: Ex. 4.2, 4.4

Solutions

Reading for Week 6: T&K: p. 135-147, 169-174, 177-195

Problem set 5: Due Thursday, February 12

T&K

p. 127 – 135: Ex. 4.8, Pr. 4.5, 4.8, 4.12

p. 148 – 151: Ex. 5.4, Pr. 5.1, 5.3, 5.4

p. 183 - 184: Ex. 8.1, 8.3; Pr. 8.2, 8.3

p. 195 - 197: Ex. 9.1, 9.3

Solutions

Reading for Week 7: T&K: p. 199 - 142

Problem set 6: Due Thursday, February 19

T&K

p. 195 – 197: Pr. 9.1, 9.5

p. 208 – 215: Ex. 1.1, Pr. 1.2, 1.8

p. 228 – 234: Ex. 2.8, Pr. 2.4 (a,b)

p. 243 – 245: Ex. 3.2, 3.4

Solutions

Reading for Week 8: T&K: p. 245 – 250, 267 – 274

Problem set 7: Due Thursday, February 26

T&K

p. 243-245: Pr. 3.1, 3.3

p. 254-258: Ex. 4.1, 4.2, Pr. 4.1, 4.2, 4.4

p. 274-279: Ex. 1.4, 1.9, Pr. 1.7

Solutions

Reading for Week 9: T&K: p. 279 - 285, 290 – 294, 297 - 303

Problem set 8: Due Thursday, March 5

T&K

p. 274 - 279: Pr. 1.2

p. 286 - 290: Ex. 2.1, Pr. 2.4, 2.5, 2.8

Solutions

Reading for Week 10: T&K: 304 - 308

Problem set 9 (last one): Due Thursday, March 12

T&K

p. 294 – 297: Ex. 3.8, Pr. 3.9

p. 308 – 311: Ex. 4.2, 4.5, Pr. 4.2, 4.5, 4.6, 4.8

Solutions