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

Syllabus - List of possible Projects - Projects webpage

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:

On Random Walks and Markov Chains / Solutions

Practice for Midterm1 / SolutionsSolution of the Midterm ;

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

Save the dates

Midterm 1: Wednesday March 12th

Spring recess: 17-23 March

Last day of Class: May 2nd

Final: Wednesday, May 7th

Office hours

Mon 2-3pm, Wed 2-3pm, Wed 5-6pm

Other possible references

General Probability:
 
[1] R. Durrett. Probability and Examples, 2nd ed. Duxbury Press, 1996.
[2] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. I. Wiley. 
 
Stochastic Processes:
 
[1] D. Williams Probability with martingales. Cambridge University Press 1991 
[2] R. Durrett Essentials of stochastic processes. Springer 1999 
 
  • Quentin Berger
  • USC, Department of Mathematics
  • 3620 S. Vermont Ave, KAP 108
  • Los Angeles, California 90089-2532