Instructor: John Pike
Teaching Assistant: Mathav Murugan
This is the first half of a year-long introduction to probability theory at the graduate level. We will begin by introducing some of the fundamental concepts (e.g. probability spaces, random variables, expectation) from a measure-theoretic perspective. Basic knowledge of abstract measure theory is assumed, but we will endeavor to keep the course as self-contained as possible. We will then discuss the notion of independence and move on to derive the laws of large numbers and some related results. Next we will introduce characteristic functions and weak convergence in order to prove the central limit theorem and various complements/extensions thereof. We will conclude with a look at stopping times, random walk, and conditional expectation.
The official course textbook is
We will follow Durrett's book fairly closely, covering most of the material from chapters 1-4.
Some supplemental texts which may be of use include:
* The book by Ross is written for an undergraduate audience, but it has many wonderful examples and may help to motivate some of the material discussed in this course.
The best way to learn this material is by working out a variety of problems on your own. Accordingly, your course grade will be primarily determined by weekly homework assignments. (There may be some small participation or presentation component as well.) Typically, new assignments will be posted on Wednesdays and will be due in class on Friday of the following week. Each student is granted two free passes to turn in homework up to a week after the posted due date. Beyond this, late work will not be accepted without a compelling reason.
You are encouraged to discuss the homework problems with your classmates, but each student must turn in their own solutions. Also, though it is perfectly acceptable to consult outside resources for help on occasion, you should spend a reasonable amount of time thinking about the problem and attempting your own solution before doing so, and you are required to explicitly cite all sources other than the official text.
Finally, please make sure that your work is clearly presented, organized, stapled, etc... It is recommended (but not required) for students to write up their assignments using TeX. In addition to making life easier on Mathav, most of you will need to learn TeX at some point and it is useful to have electronic copies of your work for later reference.