P R O B A B I L I T Y, F A L L 2 0 1 9
Lectures: Monday and Wednesday, 2.00pm3.15pm, in Warren Weaver Hall 101.
Lecturer: Paul Bourgade, office hours Thursday 23pm, 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 Christopher Thornett (thornett@cims.nyu.edu). He will run recitation sections
in Warren Weaver 101 on Fridays, 12.301.45 pm and 2.00pm3.15pm.
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 multivariable 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 Christopher Thornett at the beginning of his first recitation session, at 12.30).
Grading: problem sets (40%), midterm (20%) and a final exam (40%).
A tentative schedule for this course is:
Sep. 4.  Introduction, some combinatorics. Reading: Sections 1.1, 1.2, 1.3 and 1.4
(First occurence of probability words in History.) 
Sep. 9.  More combinatorics. Sample space and events. Reading: Sections 1.5, 2.1, and 2.2
(Kolmogorov's foundations of probability theory) 
Sep. 11.  Axioms of probability, inclusionexclusion formula. Reading: Sections 2.3 and 2.4 (try to make rigorous the solution to Lewis Carroll's famous Chelsea pensioners problem) 
Sep. 16.  Some probability distributions on finite sets. Reading: Sections 2.4 and 2.5 (example of bad use of uniform probability) 
Sep. 18.  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. 
Sep. 23.  Bayes' formula (Example). Reading: Section 3.3.

Sep. 25.  Independence (as explained by Marc Kac). Reading: Section 3.4 
Sep. 30.  Discrete random variables and their expectation. Reading: Sections 4.1, 4.2, 4.3 and 4.4. 
Oct. 2.  Discrete random variables, their variance and examples. Reading: Sections 4.5, 4.6. 
Oct. 7.  Poisson random variables and Poisson process. Reading: Section 4.7, 9.1. 
Oct. 9.  More on discrete random variables. Reading: Section 4.8, 4.9. 
Oct. 14.  No course on this Monday, course on Tuesday instead. 
Oct. 15.  Review. 
Oct 16.  Midterm exam 
Oct. 21.  Review of midterm exam and multivariate calculus. 
Oct. 23.  Continuous random variables, distribution and density. Reading: Section 5.1. 
Oct. 28.  Expectation, variance, transformation and Jacobian. Reading: Section 5.2. 
Oct. 30.  Some continuous distributions. Reading: Sections 5.3, 5.4, 5.5. 
Nov. 4.  Jointly distributed random variables. Reading: Section 6.1. 
Nov. 6.  Expectation, covariance, transformation and Jacobian. Reading: Sections 6.3, 6.7, 7.2, 7.3, 7.4. 
Nov. 11.  Conditional probability. Reading: Sections 6.4, 6.5, 7.5. 
Nov. 13.  Independence. Reading: Section 6.2. 
Nov. 18.  Moment generating functions. Reading: Section 7.7. 
Nov. 20.  Law of large numbers I. Reading: Section 8.2. 
Nov. 25.  Law of large numbers II. Reading: Section 8.4. 
Nov. 27.  Thanksgiving recess. 
Dec. 2.  Central limit theorem I. Reading: Section 8.3. 
Dec. 4.  Central limit theorem II. Reading: Section 8.3. 
Dec. 9.  Markov Chains. 
Dec. 11.  Review. 
Dec. 16.  Final exam, from 2 till 3.50pm, room 101. 