P R O B A B I L I T Y, F A L L 2 0 1 5

**Lectures**: Wednesday, 5.10pm-7pm, in Warren Weaver Hall 1302.

** Lecturer**: Paul Bourgade, office hours Thursday 10-11am, you also can email me (bourgade@cims.nyu.edu)
to set up an appointment or just drop by (Warren Weaver Hall 603).

** Course assistant**: Alexisz Gaal (gaal@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 (20%) and a final exam (40%).

A tentative schedule for this course is:

- Sep. 2. Introduction: some aspects of the random walk
- Sep. 9. Axioms of probability. Countable space: inclusion-exclusion.
- Sep. 16. Countable space: conditional probability and independence, random variables.
- Sep. 23. Probability measure on ℝ. Random variables and integration with respect to a probability measure.
- Sep. 30. Independent random variables, probability distributions on ℝ
^{n}. - Oct. 7. Sums of independent random variables, Gaussian vectors.
- Oct. 14. Convergence types.
- Oct. 21. Midterm.
- Oct. 28. Characteristic functions and central limit theorem.
- Nov. 4. Law of large numbers.
- Nov. 11. Martingales I.
- Nov. 18. Martingales II.
- Nov. 25. Thanksgiving.
- Dec. 2. Markov chains I.
- Dec. 9. Markov chains II.

- Problem set 1.
- Problem set 2.
- Problem set 3.
- Problem set 4.
- Problem set 5.
- Midterm practice.
- Midterm exam.
- Problem set 6.
- Problem set 7.
- Problem set 8.
- Problem set 9.
- Final practice.
- Final exam.