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 ℝ
^{n}. - 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 set 1. Tex file. Solutions
- Problem set 2. Tex file. Solutions
- Problem set 3. Tex file. Solutions
- Problem set 4. Tex file. Solutions
- Problem set 5. Tex file. Solutions
- Problem set 6. Tex file. Solutions
- Problem set 7. Tex file. Solutions
- Problem set 8. Tex file. Solutions
- Problem set 9. Tex file. Solutions
- Problem set 10. Tex file. Solutions
- Problem set 11. Tex file.
- Problem set 12. Tex file. Solutions
- Problem set 13. Tex file. Solutions