Probability

P R O B A B I L I T Y, S p r i n g 2 0 2 0

Lectures: Wednesday, 7.10pm-9.00pm, in Warren Weaver Hall 202.

Lecturer: Paul Bourgade, office hours Thursday 1.30-2.30pm, you also can email me (bourgade@cims.nyu.edu) to set up an appointment or just drop by (WWH 603).

Course description: A rigorous introduction to probability theory, including: discrete and continuous random variables, distribution and density functions in arbitrary dimension, conditional probability, generating functions; weak and strong laws of large numbers, the central limit theorem, random walks, discrete martingales.

Prerequisites: the course will build on infinite series, multivariable calculus, basics about linear algebra, and along the way we will introduce the required notions about set theory and elementary measure theory.

Textbooks: Our reference text will be Probability Essentials, by Jacod-Protter.

Homework: Every Wednesday for the next Wednesday.

Grading: problem sets (35%), midterm (15%) and final (50%).

A tentative schedule for this course is:

Jan. 29. Introduction: some aspects of the random walk
Feb. 5. Axioms of probability. Countable space: inclusion-exclusion.
Feb. 12. Countable space: conditional probability and independence, random variables.
Feb. 19. Probability measure on ℝ. Random variables and integration with respect to a probability measure.
Feb. 26. Independent random variables, probability distributions on ℝⁿ.
Mar. 4. Sums of independent random variables, Gaussian vectors.
Mar. 11. Characteristic functions.
Mar. 18. Spring break.
Mar. 25. Midterm.
Apr. 1. Convergence types and central limit theorem.
Apr. 8. Law of large numbers.
Apr. 15. Martingales I.
Apr. 22. Martingales II.
Apr. 29. Markov chains I.
May. 6. Markov chains II.
May 13. Final.

Problem sets.