MATH40480

Probability theory has its roots in games of chance, such as coin tosses or throwing dice. By playing these games, one develops some probabilistic intuition. Such intuition guided the early development of probability theory and allowed for rigorous mathematical statements to be made concerning such games as well as more complex problems involving randomness.

In this course, we will develop the mathematical tools required for the study of randomness. Measure theory and integration play a fundamental role in this development, as does independence and various notions of convergence of random variables. Having covered the basics of probability and measure, we will learn about (and prove in some cases) some of the most essential results in probability theory such as laws of large numbers, Borel-Cantelli lemmas, and central limit theorem. We will also introduce and study conditional expectations, martingales and Brownian motion.