Homepage of MATH 180B

Introduction to Probability

Winter 2006

Department of Mathematics

University of California, San Diego

COURSE INFORMATION SHEET

Instructor Contact Information is posted on her homepage

Teaching Assistant

…      Arthur Berg, aberg@math.ucsd.edu

…      Office Hours: Thursdays 1:30pm-3pm, AP&M 2325

…      Finals Week Office Hour: Tuesday, 3pm-4pm, AP&M 2325

LECTURES

…      Week 1:  Pitman, 5.R, 6.1, & 6.2, where R=Review.

…      Week 2:  Pitman, 6.3 except examples 3-5, i.e., no mixing of continuous and discrete for now, & 6.4 through example 3.

…      Week 3:  Pitman, 6.5 except the very last section on several independent normal random variable & Taylor and Karlin, III.1 & III.2.

…      Week 4:  Exam 1 (covers Pitman Chapter 5 Review and Chapter 6, plus Taylor and Karlin III.1; click here for a list of distributions/densities that you should have committed to memory); Taylor and Karlin III.3.

…      Week 5:  Taylor and Karlin, III.4, III.5, and III.6.

…      Week 6:  Taylor and Karlin, III.8, III.9, and IV.1. Regular Matrices Maple Demo.

…      Week 7:  Taylor and Karlin, IV.2, and IV.3.

…      Week 8:  Exam 2 (covers Taylor and Karlin Chapters III through IV.2).  Taylor and Karlin IV.3 and IV.4.

…      Week 9:  Taylor and Karlin V.1, V.2, V.3

…      Week 10:  Taylor and Karlin V.4, V.5, and V.6

…      Final Exam:  Will be held in Peterson Hall Room 104

HOMEWORK

…      Assignment 1:  Due Friday, 1/13; Pitman, 5.R.2, 5.R.5, 5.R.6, 5.R.7, 5.R.8, 5.R.10c, & 5.R.12, where R=Review.

…      Assignment 2:  Due Friday, 1/20; Pitman, 6.1.1, 6.1.2, 6.1.3, 6.2.2, 6.2.4, & 6.2.10.

…      Assignment 3:  Due Friday, 1/27: Pitman, 6.3.4, 6.3.5, 6.3.12, 6.4.1, 6.4.4, 6.5.2, 6.5.4, & 6.5.5.

…      Bonus Problem: 6.5.3, where the rules are that the bonus problem must be solved using only your own brain and the Pitman book, i.e., not consulting myself, the TA, or other resources.  You may collaborate with your classmates, in which case the points will be shared equally among the collaborators.  In order to get credit, you or you and your collaborators must present your solution to me in my office hours, or by appointment by Monday, January 30.  Note: You may not share your ideas/work with anyone in the course, except those identified as collaborators.

…      Assignment 4:  Due Friday 2/3: Taylor and Karlin, III.1 Exercise 1.3 and Problems 1.1 and 1.3 & III.2 Exercise 2.6 and Problems 2.1 and 2.4.

…      Assignment 5:  Due Friday 2/10: Taylor and Karlin, III.3 Exercises 3.4 and 3.5 and Problems 3.2, 3.3, and 3.4 & III.4 Exercises 4.8 and 4.9 and Problems 4.8, 4.14, and 4.16.

…      Assignment 6:  Due Friday 2/17:

1.   Taylor and Karlin, Section III.5 Exercise 5.2, 5.5, 5.7, 5.8 and 5.9 and Problems 5.2 and 5.4 and Section III.6 Exercise 6.3 and 6.4 page 167.

2.   For the Gamblerνs Ruin problem with a total of $N and p_i=q_i=1/2, use the method of difference equations to derive a formula for the probability of ruin starting from $1.  What can you conclude about the probability of ruin when playing an infinitely rich adversary? Using (6.7) in III.6.1, what can be said about the expected time to ruin when playing a fair game against an infinitely rich adversary?

3.   For the Gamblerνs Ruin problem with a total of $N, p_i=p, q_i=q, and p not equal to q, use the method of difference equations to derive a formula for the expected length of the game when the initial wealth of player A is $1.  What can you conclude about the expected time until ruin when playing an infinitely rich adversary?

…      Assignment 7:  Due Friday 2/24:  Taylor and Karlin III.8 Exercises 8.2 and 8.4 and Problems 8.1, 8.2, and 8.3.  Taylor and Karlin III.9 Exercises 9.2 and 9.3 and Problems 9.2, 9.5, and 9.9.  Taylor and Karlin IV.1 Exercises 1.5 and 1.10 and Problems 1.1, 1.3, 1.5, 1.6, and 1.13.  Taylor and Karlin IV.2 Exercise 2.6 and Problems 2.1 and 2.4.

…      Bonus Problem:  Show that the generating function of the offspring distribution has two fixed points in [0,1] if and only if the derivative is greater than 1 at 1, where the rules are the same as in assignment 3.  Due by Monday 2/27.

…      Assignment 8:  Due Friday 3/10:

1.   Modify Taylor and Karlin IV.3 Exercise 3.1 by adding two states 8 and 9 and the making following changes to the transition matrix: P_{0i}=1/2 for i=1, 8, P_{89}=1, and P_{90}=1.  Find an integer N such that n>=N implies P_{ii}^{(n)}>0 for all i.

2.   Taylor and Karlin IV.3 Exercise 3.3 and Problems 3.2 and 3.3.

3.   Consider an irreducible and positive recurrent Markov chain.  Let r_i be the expected number of visits to i after time 0 that occur at or before the first return to 0 starting from state 0.  Show that

1.   r_0=1

2.   r_i=sum_j r_jp_{ji} for all i.

3.   r_i is positive and finite for all i.

4.   Taylor and Karlin IV.4. Exercise 4.3 and Problems 4.7 and 4.8.

…      Assignment 9:  Due Friday 3/17:

1.   Taylor and Karlin V.1 Exercises 1.3, 1.6, and 1.7 and Problems 1.1 and 1.9.

2.   Taylor and Karlin V.2 Exercises 2.3 and 2.4 and Problems 2.1, 2.4, and 2.7.

3.   Taylor and Karlin V.3 Exercises 3.6 and 3.7 and Problems 3.1-3.4, 3.6 and 3.7.  I basically did 7 in class, but note that P(W_{k+1}<=t)=P(N((0,1])>=k+1).  In class I had a k rather than a k+1, so be sure the check the inequalities for correctness.

…      Recommended

1.   Pitman 4.2.5, 4.2.6, 4.2.15, 4.2.16. (Doing these will reinforce the characterization of a Poisson process in terms of independent exponential lambda sojourn times.  Note that what T&K label as W, P labels as T, and what T&K label S, P labels as W.)

2.   Problems from Taylor and Karlin V.4 Exercises 4.4 and 4.5 and Problems 4.6, 4.8, 4.9, and 4.10.