P R O B A B I L I T Y, F A L L 2 0 1 4

**Lectures**: Wednesday, 5.10pm-7pm, in Warren Weaver Hall 101.

** Lecturer**: Paul Bourgade, office hours Thursday 10.30-12pm, you also can email me (bourgade@cims.nyu.edu)
to set up an appointment or just drop by (Warren Weaver Hall 603).

** Course assistant**: Insuk Seo (insuk@cims.nyu.edu).

**Course description**:
The course introduces the basic concepts and methods of probability. Topics include: probability spaces, random variables, distributions, law of large numbers, central limit theorem, random walk, Markov chains and martingales in discrete time, and if time allows diffusion processes including Brownian motion.

**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 (40%), midterm (30%) and a final project (30%).

A tentative schedule for this course is:

- Sep. 3. Introduction: some aspects of the random walk
- Sep. 10. Axioms of probability. Countable space: inclusion-exclusion.
- Sep. 17. Countable space: conditional probability and independence, random variables.
- Sep. 24. Probability measure on ℝ. Random variables and integration with respect to a probability measure.
- Oct. 1. Independent random variables, probability distributions on ℝ
^{n}. - Oct. 8. Sums of independent random variables, Gaussian vectors.
- Oct. 15. Convergence types.
- Oct. 22. Characteristic functions and central limit theorem.
- Oct. 29. Law of large numbers.
- Nov. 5. Midterm.
- Nov. 12. Martingales I.
- Nov. 19. Martingales II.
- Nov. 26. Markov chains I.
- Dec. 3. Markov chains II.
- Week of Dec. 10. Students talks