Math 505b: Applied Probability
MondayWednesdayFriday, 11.50pm, in KAP 148
Email: qberger@usc.edu
Textbook: Gimmett and Stirzaker: Probability & Random Processes (3rd).
Grader: Jia Zhuo, jiazhuo@usc.edu, KAP 248A
Syllabus  List of possible Projects  Projects webpage
Review session: Friday, May 2nd, 1pm1:50pm KAP 148, then 2pm3:20pm 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:
On Random Walks and Markov Chains / Solutions;
Practice for Midterm1 / Solutions; Solution of the Midterm ;
Lecture notes:ChI.RandomWalk.pdfChII.Markov.Chains.Introduction.pdfChII.Markov.Chain.Stationary.distribution.pdf ChIII.Martingales.Introduction.Stopping.Times.pdfChIII.Martingales.Convergence.Theorems.pdfChIV.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 nonnegative up to time 2n is the same as the probability that the random walk is equal to 0 at time 2nBallot 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, 6check 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, 5Show that the WrightFisher model (HW2) is a martingale. Deduce the probability that allele A wins over allele B, from the (not yet) nonrigorous 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 900892532
 Phone: (213) 821  1628
 Email: qberger@usc.edu