Math 561, Probability Theory
Spring 2013

Course Information

Professor:Kay Kirkpatrick
Office:334 Illini Hall
Contact:kkirkpat(at)illinois.edu
Website:https://faculty.math.illinois.edu/~kkirkpat/561G1.html
Lectures: MWF 3:00-3:50 (141 Altgeld Hall)
Office hours: Mondays and Wednesdays, 2:00-2:50pm, or by appointment. I would be happy to answer your questions in my office anytime as long as I'm not otherwise engaged, and before and after class are good times to catch me either in my office or in the classroom.
Textbook: R. Durrett: Probability: Theory and Examples (3rd edition) Duxbury Press, 2005. (It is okay to use another edition for studying, but the homework problems will be from the 3rd edition, and you can check my textbook or a classmate's to make sure you're doing the right problems.)
Prerequisite: Math 540 Real Analysis I. We will review measure theory topics as needed. (Math 541 is nice to have, but not necessary.)
Grading policy: Homework: 50% of the course grade
Final Exam: 50%
Here's an old (take-home) final for practice. Our final will be in class, Friday, May 3, from 1:30-4:30pm in AH 141.
Homework (due Fridays in class): You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately. HW #1, due Friday, January 18, in class: the info sheet.

HW #2: to be announced in class.

Topics:

This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics: (1) the basic limit theorems (the law of large numbers, the central limit theorem and the large deviation principle); (2) martingales and their applications. If time allows, we will give a brief introduction to Brownian motion. The prerequisite for Math 561 is Math 540.

Why study probability?

Two main reasons: uncertainty and complexity. Uncertainty is all around us and is usefully modeled as randomness: it appears in call centers, electronic circuits, quantum mechanics, medical treatment, epidemics, financial investments, insurance, games (both sports and gambling), online search engines, for starters. Probability is a good way of quantifying and discussing what we know about uncertain things, and then making decisions or estimating outcomes. Some things are too complex to be analyzed exactly (like weather, the brain, social science), and probability is a useful way of reducing the complexity and providing approximations. And the reason I study probability: statistical mechanics, which combines both the uncertainty of quantum mechanics, and the complexity of zillions of particles interacting.