Problem Sets for Math 180c, Spring 2010

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

Reading for Weeks 1 & 2: Goodman – Chapters 7 & 8

Optional reading: Lawler: p. 65-82 and Durrett: p. 158-181

Problem set 1: Due Thursday April 8

Goodman Chapter 7 (p. 197-199) : 1, 2, 4 [slightly misstated; all inequalities should be strict], 6, 11

Comments: In Problem 11 above, E[N|T] is the random variable (RV) which at time t is the expected value of the discrete RV with probability P[N=n|T=t]. See Sec. 5.2 & 5.3.)

Goodman Chapter 8 (p. 219-222): 1, 6, 8, 13

Comments: Problem 13 involves an interesting calculation and requires some care. It is NOT an M/M/1 queue. I recommend that you do it on a spreadsheet.

Reading for Week 3: Goodman – Chapter 9 and pp. 241 – 244 of Chapter 10.

Problem set 2: Due Thursday April 15

Goodman Chapter 8 (p. 222): Exercise 14

Explanation: In (a) of this problem, the notation x(2) means the state of the system at the time of the second jump, x(3) means the state of the system at the time of the third jump, and similarly for x(4). The system starts at x(0) = 0 and must have x(1) = 1, since the first jump occurs when the first customer arrives.

Goodman Chapter 9 (p. 238-240): Exercises 1, 2, 4, 6, 7

Also:

1. Men and women enter a supermarket according to independent Poisson processes X and Y having respective rates a and b respectively. For any integer k > 0, what is the probability that exactly k men will enter the store before the first women enters? Explain your answer briefly.

2. Suppose that X and Y are independent uniform random variables on [0,2] and [0,3] respectively. Find (a) P[X < Y] by evaluating an appropriate integral, and (b) the distribution of the random variable Z = Min(X,Y). Show your calculations.

Problem set 3 (a short one): Due Thursday April 22

Goodman Chapter 7 (pp. 197-199) : 3, 9

Goodman Chapter 10 (pp. 156-259): 2 (parts a. and b. only )

Reminder: First midterm Wednesday April 21 in class. See Rules.

Reading for Week 5: Goodman- all of the rest of Chapter 10, but skip the proof of Example 10.2 (i.e. bottom to p. 245 to top of p. 248).

Taylor and Karlin – pp. 380-384

Problem set 4 (another short one): Due Thursday April 29

Goodman Chapter 9 (p. 240) : #8

Goodman Chapter 10 (p. 256-259) #3, #4, #7

Reading for Week 6: Goodman pp. 261-280

Problem set 5: Due Thursday May 6

Taylor and Karlin: p. 393 Problem 5.2

Misprint in Problem 5.2: the index 4 is missing on the Greek letter “mu.”

Goodman Chapter 10 (p. 258): #10, #11, #12, #13, #14, #15

Correction for #15: It should read “the probability is 1/(n+1) that the customer enters the system.” Here n = 0, 1, 2, …

Goodman Chapter 11 (p. 291): #1, #3

Reading for Week 6: Goodman pp. 280 – 283 (We may not get through all of this.)

Problem set 6: Due Thursday May 13

Goodman Chapter 11 (pp. 291-296): Exercises #4, #5, #6, #7, #8, #9, #11, #12, #13

Reminder: Second midterm Wednesday May 19 in class. See rules.

Reading for Week 7: Goodman pp. 299-309.

Problem set 7 (a short one): Due Thursday, May 20

Goodman Chapter 11 (pp. 291-296): Exercises #14, #23 (a) only

Goodman Chapter 12 (pp. 329-335): Exercises #1 (a,b) only, # 5.

Reading for Week 9: Goodman pp. 209-329.

Problem set 8: Due Thursday, May 27

Goodman Chapter 12 (pp. 329-335): Exercises #3, #7, #8, #11, #12, #14

Reading for Week 10: Taylor and Karlin pp 473-483

Further suggested reading on Brownian motion: Lawler pp. 173-176, Durrett pp 242-246

Problem set 9 (the last one!): Due Friday, June 4 (anytime)

Goodman Chapter 12 (pp. 329-335): Exercises #15, #17, #19, #20, #21, #24, #28, #30

Taylor and Karlin: p. 487 Exercise 1.1 and p. 488 Problem 1.1.