P R O B A B I L I T Y,    S P R I N G  2 0 1 7

Lectures: Tuesday and Thursday, 9.30am-10.45am, in Warren Weaver Hall 101.

Lecturer: Paul Bourgade, office hours Wednesday 2-3pm, you also can email me (bourgade@cims.nyu.edu) to set up an appointment or just drop by (Warren Weaver Hall 603).

Course assistant: The teaching assistant is Zhe Wang (zhe.wang.frank@gmail.com). He will run recitation sections in Warren Weaver 101 on Fridays, 3.30pm-4.45pm.

Course description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.

Prerequisites: This course is intended for math majors and other students with a strong interest in mathematics. It requires fluency in topics such as multi-variable integration.

Textbook: Our reference text will be A first course in probability, by Sheldon Ross.

Homework: Posted on this page every Thursday for Friday the week after (you should then give it to Zhe Wang during the recitation session).

Grading: problem sets (40%), midterm (20%) and a final exam (40%).

A tentative schedule for this course is:

Jan. 24.Introduction, some combinatorics. Reading: Sections 1.1, 1.2, 1.3 and 1.4
(Permutations and Combinations in the 6th century B.C., A conversation between Fermat and Pascal)
Jan. 26.More combinatorics. Sample space and events. Reading: Sections 1.5, 2.1, and 2.2
(Kolmogorov's foundations of probability theory)
Jan. 31. Axioms of probability. Reading: Sections 2.3 and 2.4
Feb. 2. Some probability distributions on finite sets, Inclusion-exclusion. Reading: Sections 2.4 and 2.5.
Feb. 7.Conditional probability. Reading: Sections 3.1, 3.2 and 3.5.
Sometimes conditioning changes it all, sometimes conditioning does not matter: keep the order of magnitude in mind.
Feb. 9.Bayes' formula (Example). Reading: Section 3.3.
Feb. 14.Independence (as explained by Marc Kac). Reading: Section 3.4
Feb. 16.Discrete random variables. Reading: Sections 4.1 and 4.2.
Feb. 21. Expectation, variance. Reading: Sections 4.3, 4.4 and 4.5.
Feb. 23.Some discrete distributions I. Reading: Section 4.6
Feb. 28.Some discrete distributions II. Reading: Section 4.7.
Mar. 2.Some discrete distributions III. Reading: Section 4.8.
Mar. 7.Review
Mar. 9.Midterm exam
Mar. 14.Spring Break
Mar. 16.Spring Break
Mar. 21.Review of midterm exam and multivariate calculus.
Mar. 23.Continuous random variables, distribution and density. Reading: Section 5.1.
Mar. 28.Expectation, variance, transformation and Jacobian. Reading: Section 5.2.
Mar. 30.Some continuous distributions. Reading: Sections 5.3, 5.4, 5.5.
Apr. 4.Jointly distributed random variables. Reading: Section 6.1.
Apr. 6.Expectation, covariance, transformation and Jacobian. Reading: Sections 6.3, 6.7, 7.2, 7.3, 7.4
Apr 11.Conditional probability. Reading: Sections 6.4, 6.5, 7.5
Apr. 13.Independence. Reading: Section 6.2.
Apr. 18.Moment generating functions. Reading: Section 7.7.
Apr. 20.Law of large numbers I. Reading: Section 8.2.
Apr. 25.Law of large numbers II. Reading: Section 8.4.
Apr 27.Central limit theorem I. Reading: Section 8.3.
May 2.Central limit theorem II. Reading: Section 8.3.
May 4.Markov Chains.
May 10 to 16.Final exam.

Problem sets.