Probability

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 ℝⁿ.
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 sets.

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.