Theory of Probability

Course:MATH-UA.233 Theory of Probability, Fall 2021
Time: Monday, Wednesday 2:00 - 3:15 PM
Room: TBA at Courant Institute / Warren Weaver Hall
Instructor: Professor Yuri Bakhtin, contact info
Office hours: Tentatively Tuesday, 2-3pm Wednesday 3:30-4:30, CIMS/WWH office 713
Recitation: TBA
Course description:

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains, applications.

Textbook: A first course in probability, by Sheldon Ross.
Prerequisites: This course is intended for math majors and other students with a strong interest in mathematics. It requires fluency in calculus topics such as series, derivatives, (multi-variable) integration. It makes sense to look through the textbook in advance to know what to expect.
Homework: Problem sets will be posted online. Solutions are usually to be submitted for grading every week. Late homework will not be accepted.
Exams: Two midterm exams and the final exam are in-class and closed book.
Grading: Two lowest weekly homework scores will be dropped. The total homework score counts for 30% of the course grade. Each midterm counts for 15% of the course grade. The final exam counts for 40% of the course grade.
Tentative schedule: The semester begins on Thursday Sep 2, so no classes in the first week
Sep 6 Labor Day, no class
Sep 8 Introduction, some combinatorics. Reading: Sections 1.1, 1.2, 1.3, 1.4
Sep 13 More combinatorics. Sample space and events. Reading: Sections 1.5, 2.1, 2.2
Sep 15 Axioms of probability, inclusion-exclusion formula. Reading: Sections 2.3, 2.4
Sep 20 Some probability distributions on finite sets. Reading: Sections 2.4, 2.5
Sep 22 Conditional probability. Reading: Sections 3.1, 3.2, 3.5
Sep 27 Bayes' formula. Reading: Section 3.3.
Sep 29 Independence. Reading: Section 3.4
Oct 4 Discrete random variables and their expectation. Reading: Sections 4.1, 4.2, 4.3, 4.4.
Oct 6 Midterm exam
Oct 11 The class is moved to Tuesday Oct 12(Fall Break/Legislative Monday)
Oct 12 Discrete random variables, their variance and examples. Reading: Sections 4.5, 4.6
Oct 13 Poisson random variables and Poisson process. Reading: Section 4.7, 9.1
Oct 18 More on discrete random variables. Reading: Section 4.8, 4.9
Oct 20 Review of calculus. Continuous random variables, distribution and density. Reading: Section 5.1
Oct 25 Expectation, variance, transformation and Jacobian. Reading: Section 5.2
Oct 27 Some continuous distributions. Reading: Sections 5.3, 5.4, 5.5
Nov 1 Jointly distributed random variables. Reading: Section 6.1
Nov 3 Expectation, covariance, transformation and Jacobian. Reading: Sections 6.3, 6.7, 7.2, 7.3, 7.4
Nov 8 Conditional probability. Reading: Sections 6.4, 6.5, 7.5
Nov 10 Independence. Reading: Section 6.2
Nov 15 Review
Nov 17 Midterm exam
Nov 22 Moment generating functions. Reading: Section 7.7.
Nov 24 Law of large numbers I. Reading: Section 8.2.
Nov 29 Law of large numbers II. Reading: Section 8.4.
Dec 1 Central limit theorem I. Reading: Section 8.3.
Dec 6 Central limit theorem II. Reading: Section 8.3.
Dec 8 Markov Chains.
Dec 13 Review
TBA Final exam