MAT/STA 235A: Probability Theory (Fall 2025) (CRN 36578/47522)
     2:10-3 AM, 140 Physics

http://www.math.ucdavis.edu/~gravner/MAT235A/
INSTRUCTOR:
TA:

PREREQUISITES: A solid working knowledge of advanced calculus. Measure theory will be reviewed at the beginning of the course, but some additional reading would be advisable if you have never seen it before. Here are the topics from measure theory: measurable spaces, Lebesgue integral, Lebesgue measure in Rd, monotone and dominant convergence theorems, Fubini's theorem, and Radon-Nikodym theorem. Also, knowledge of basic combinatorics, such as permutations, combinations, selections with and without replacement.

TEXTBOOK: The required textbook is R. Durrett, Probability: Theory and Examples (5th edition, Cambridge University Press, 2019. Free online version is available from the author. Sections 1.1-1.7, 2.1-2.6, 3.2-3.4, 3.6 will be covered in this part of the course.

There are many other books that cover this material, e.g., L. Breiman, Probability (1968), D. Williams, Probability With Martingales (1991), J. Jacod, P. Protter, Probability Essentials (2000). The last two books are similar in scope (covering most of 235AB) and easiest to read, thus suitable as gentler supplements to Durrett.

TOPICS:

  • introduction: probability measures, random variables, expected values, independence;
  • strong and weak laws of large numbers;
  • weak convergence and central limit theorem; and,
  • time permitting: Poisson convergence, large deviations.

    • GRADE: Course grade will be based on the following:

    ADDITIONAL POLICIES:

    Homework will be assigned from time to time but will not be collected. You are encouraged to solve all problems on your own.

    The midterm and final exams will be take-home. Midterm will be assigned at the beginning of November and Final at the beginning of December (last week of classes). On the exams you must work alone and use only your notes from this class. Late submissions of your work will not be accepted.

    Your solutions will be graded not only on correctness, but also on clarity, organization, and quality of writing.

    There are free resources available on the web. A good example are the Probability Tutorials. Freely available (and highly regarded) books on measure theory include Measure, Integration & Real Analysis by Sheldon Axler and An Introduction to Measure Theory by Terence Tao. You may also use my undergraduate lecture notes to brush up on elementary probability.