Advanced probability and applications
Weekly outline

"Probability theory is nothing but common sense reduced to calculation." PierreSimon de Laplace, 1812 (see other interesting quotes from PierreSimon de Laplace)
Lectures:
 In person in room CM 1 120 on Wed 24 PM and in room INR 219 on Thu 810 AMExercise sessions:
 In person in room INR 219 on Thu 1012 AM
Grading scheme:
 Graded homeworks 20%
 Midterm 20%
 Final exam 60%Principle for the graded homeworks: each week, one exercise is starred and worth 2% of the final grade; the best 10 homeworks (out of 12) are considered. The homework is due on Wednesday of the following week, in lecture or by 5pm in the dropbox outside of INR 131.
Midterm exam: Thursday, April 11 8:15am10am, CM 1 3
 Allowed material: one cheat sheet (i.e., two singlesided A4 handwritten pages)
Final exam: Thursday 20.06.2024 from 09h15 to 12h15 (CE1).
 Allowed material: two cheat sheets (i.e., four singlesided A4 handwritten pages)
Course instructor:
 Please note that the content of the exam will focus more on the second part of the course (but also on the first part)
Prof. Yanina Shkel  MIL  IPG  INR 131  yanina.shkel@epfl.ch
Teaching assistants:Marco Bondaschi  LINX  IPG  INR 136  marco.bondaschi@epfl.chAnuj Yadav  MIL  IPG  INR 034  anuj.yadav@epfl.chCourse Webpage:References:
Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability, 1st edition, www.ProbabilityBookstore.com, 2007.
Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory, 2nd edition, World Scientific, 2006.
Geoffrey R. Grimmett, David R. Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.
Geoffrey R. Grimmett, David R. Stirzaker, One Thousand Exercises in Probability, 1st Edition, Oxford University Press, 2001.
Sheldon M. Ross, Stochastic Processes, 2nd edition, Wiley, 1996.
William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1&2, Wiley, 1950.
(more advanced) Richard Durrett, Probability: Theory and Examples, 4th edition, Cambridge University Press, 2010.(more advanced) Patrick Billingsley, Probability and Measure, 3rd edition, Wiley, 1995.
Mediaspace channel for the course (please note that these videos were made for a previous version of the course taught by Olivier Lévêque: there will be some differences with this year's version)Recordings of live lectures (from 2023, Olivier Lévêque)
Minimum grade: 24/60
Maximum grade: 55/60

Wed: Sigmafields (sections 1.1, 1.2, 1.3)
Thu: Random variables (sections 1.4, 1.5), probability measures (section 2.1) 
Wed: Probability measures, Distributions of random variables (sections 2.1, 2.2, 2.3, 2.4)
Thu: The Cantor's set (Section 2.5); Independence (sections 3.1, 3.2, 3.3, 3.4) 
Guest Lecture by TAs Anuj and Marco this week!
Wed: Independence (section 3.4); Do independent random variables really exist? (section 3.5); Convolution (section 3.6)
Thu: No class! (Exercise session as usual) 
Wed: Expectation (section 4), Characteristic function (section 5)
Thu: Characteristic function (section 5), random vectors (section 6.1) 
Wed: Gaussian random vectors (sections 6.2, 6.3)
Thu: Inequalities (section 7); Convergences of sequences of random variable (sections 8.1, 8.2, 8.3)

Wed: BorelCantelli lemma; Laws of large numbers  weak and strong, proof (sections 8.3, 8.4, 8.5)
Thu: convergence of the empirical distribution; (sections 8.6)
NB: Because of the midterm next week, Hwk 6 is due on Thursday, April 17 only!

Wed: NO CLASS
Thu, 8:1510:00 AM, in room CM 13: Midterm
 content: all the course + exercises until week 6 (except hwk 6)
 allowed material: one cheat sheet (i.e., two singlesided A4 handwritten pages) 
Wed: Kolmogorov’s 01 law, StPetersburg paradox (sections 8.7, 8.8), (section 9.1)
Thu: Convergence in distribution, equivalent definitions of convergence in distribution, Lindeberg's principle (sections 9.1, 9.3, 9.4)