Ph.D. Qualifying Exam – Prob

Applied Probability (ACMS 60850): Qualifying Exam Guide

Textbook: “Probability and Random Processes” by Grimmett and Stirzaker, Third edition 2009, Oxford Univ Press, ISBN 978-019-857222-0.

Syllabus: You are responsible for these materials:

1. Basic setup of probability theory (including sample spaces, conditional probability, independence). Random variables (including the elements of measure and integration theory).
2. Discrete random variables (including random walks).
3. Continuous random variables, the basic distributions, sums of random variables.
4. Monte Carlo simulations.
5. Generating functions, branching processes, basic theory of characteristic functions.
6. Laws of large numbers, central limit theorems.
7. Markov chains (embedding, birth and death processes, Poisson processes)
8. Convergence of random variables (convergence in distribution, probability, mean-square, almost surely).
9. Various stochastic processes, including Brownian motion, and applications.
10. Martingales (discrete version only), including stopping times and optimal stopping.
11. The rudiments of stochastic integration (including Ito’s formula and the Black-Scholes differential equation).

Here are sample exams:

1. 2020 Qualifying Exam in Probability (3 hours).
2. 2020 Qualifying Exam answers (DO NOT read until you try the exam first).
3. 2018 Qualifying Exam in Probability (3 hours).
4. 2018 Qualifying Exam answers (DO NOT read until you try the exam first).
5. 2017 Qualifying Exam in Probability (3 hours).
6. 2017 Qualifying Exam answers (DO NOT read until you try the exam first).
7. A sample midterm exam (90 minutes).
8. Midterm exam answers.
9. A sample final exam (2016 Fall take home).
10. Final exam answers (2016 Fall take home).
11. A sample final exam (2017 Fall take home).
12. Final exam answers (2017 Fall take home).