# Math 505b: Applied Probability

Monday-Wednesday-Friday, 1-1.50pm, in KAP 148

Email: qberger@usc.edu

Textbook: Gimmett and Stirzaker: Probability & Random Processes (3rd).

Grader: Jia Zhuo, jiazhuo@usc.edu, KAP 248A

Review session: Friday, May 2nd, 1pm-1:50pm KAP 148, then 2pm-3:20pm KAP 163.

 Due Date Subject Correction Homework #1 Wed. January 29th HW1.pdf HW1.Solution.pdf Homework #2 Wed. February 19th HW2.pdf HW2.Solution.pdf Homework #3 Wed. February 26th HW3.pdf HW3.Solution.pdf Homework #4 Wed. March 5th HW4.pdf HW4.Solution.pdf Homework #5 Wed. March 26th HW5.pdf HW5.Solution.pdf Homework #6 Mon. April 7th HW6.pdf HW6.Solution.pdf

A collection of problems:

Problems on Martingales

Miscellaneous Problems

Lecture notes:
ChI.RandomWalk.pdf
ChII.Markov.Chains.Introduction.pdf
ChII.Markov.Chain.Stationary.distribution.pdf
ChIII.Martingales.Introduction.Stopping.Times.pdf
ChIII.Martingales.Convergence.Theorems.pdf
ChIV.Continuous.Time.Processes.pdf
• Practice Problems, week 1
p170: 2, 4, 5
• Practice problem, week 2
Random walks: prove that the probability that the random walk stays non-negative up to time 2n is the same as the probability that the random walk is equal to 0 at time 2n
Ballot Theorem: Suppose that in an election, candidate A receives a votes and candidate B receives b votes. A wins over B (a>b). Assuming random ordering of the votes, what is the probability that A is always ahead of B in the vote count?
• Practice problem, week 3
p219: 1, 2, 3, 6
check properties (2) and (3) p214 in the book.
p223: 2, 5
• Practice problem, week 4
p225, Example (6)
p225: 1, 3, 4
• Practice problem, week 5
p236: 3, 4, 6, 11
• Practice problems, week 6
Problems on Markov Chains (sheet of problems given above): pb 4, 6, 7, 8, 9, 13
• Practice problems, week 8
p475: 1, 4, 5
Show that the Wright-Fisher model (HW2) is a martingale. Deduce the probability that allele A wins over allele B, from the (not yet) non-rigorous De Moivre's approach (seen in class).
• Practice problems, April 2nd
p495: 4, 7, 8
• Practice problems, April 9th
p508: 1, 6, 7, 8, 13

• Quentin Berger
• USC, Department of Mathematics
• 3620 S. Vermont Ave, KAP 108
• Los Angeles, California 90089-2532