Essentials of Probability MATH-GA.2901 Spring 2024

Syllabus

Instructor: Professor Yuri Bakhtin

Lectures: Monday, Wednesday 4:55-6:10PM CIWW 201

Office hours: Tentatively, Tuesday 2-4pm Room: CI/WWH 729

Prerequisites: The course will routinely use a variety of tools from undergraduate calculus/analysis such as limits, series, Taylor expansions, partial derivatives, multiple integrals. We will also use basics of linear algebra and complex numbers. Modern probability theory is based on measure theory. Acquaintance with undegraduate probability and measure theory is a plus but it will not be assumed. The most important informal requirement is to be ready to study rigorous mathematics and proofs.

Books:

The official book of the course is

[W] Knowing the Odds: An Introduction to Probability (Graduate Studies in Mathematics, 139) by John Walsh, although, I am not going to follow it too closely. It is available electronically through NYU: https://www-ams-org.proxy.library.nyu.edu/books/gsm/139/gsm139.pdf

If you want to buy a paper copy, the cheapest ones I have found are on amazon.com but please do your own search.

Another good book is

[JP] Probability Essentials by Jean Jacod and Philip Protter available electronically through NYU at https://link-springer-com.proxy.library.nyu.edu/book/10.1007/978-3-642-55682-1

Course outline: This course introduces basic concepts and methods of probability theory and some applications. It is based on measure theory and is meant to be (mostly) rigorous. No prior knowledge of probability or measure theory is assumed. The plan is to cover most of the topics in the book [W], with varying level of detail: probability spaces, random variables, distributions, independence, expectations, conditional expectations, notions of convergence, Law of Large Numbers, Central Limit Theorem, characteristic functions, Markov chains, random walks, martingales. If time permits, we will study basics of the Wiener process aka Brownian motion.

Homework assignments will be collected every week or two. Most of the homework problems will come from [W]. They will count for 30% of the final score. The lowest homework score will be dropped.

Midterm exam will be given in class on Wednesday March 13. It will count for 20% of the final score.

Final exam will count for 50% of the final score.