Advanced probability and applications
Weekly outline
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"Probability theory is nothing but common sense reduced to calculation." Pierre-Simon de Laplace, 1812 (see other interesting quotes from Pierre-Simon de Laplace)
Lectures:
- In-person in the room DIA 004 on Wed 1-4 PM and ELG 120 on Thu 9-12 AM.
- Thursday lectures will take place only on select weeks. Other weeks Thursdays are reserved for the exercise session. This will be clearly marked each week on Moodle.
Exercise Sessions:
- In-person in the room ELG 120 on Thu 9-12 PM on select weeks. This will be clearly marked each week on Moodle.
- Problem sets will be posted for each unit covered in lecture. You are expected to work on them outside of class and during exercise sessions. Problem sets will not be graded. However, it is important that you do them regularly if you would like to succeed in the course.
Grading Scheme:- Midterm exam #1 - 25%
- Midterm exam #2 - 25%
- Final exam - 50%
Midterm Exam #1: Wednesday, October 8, 1:15pm - 4pm, room DIA 004 and AAC 0 08.
- Allowed material: one cheat sheet (i.e., two single-sided A4 handwritten pages).
Midterm Exam #2: Wednesday, November 26, 1:15pm - 4pm, room DIA 004 and AAC 2 31.
- Allowed material: two cheat sheets (i.e., four single-sided A4 handwritten pages). You may re-use the cheatsheet from midterm 1, and create a new additional page. Or, create two pages from scratch.
Final Exam: Wednesday, January 21, 2026, 9:15am - 12:15pm, room CM 1 121.
- Allowed material: two cheat sheets (i.e., four single-sided A4 handwritten pages).
- Please note that the exam content will focus more on the part of the course not covered on the midterms, but will also cover material already covered by the midterms.
Prof. Yanina Shkel || INR 131 || yanina.shkel@epfl.ch
Teaching and Student Assistants:Anas Himmi || anas.himmi@epfl.chPierre Fasterling || pierre.fasterling@epfl.chCourse Webpage:References:- Terence Tao, An Introduction to Measure Theory, Preprint, Softcover ISBN: 978-1-4704-6640-4
- Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability, 1st edition, 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.
- 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)-
Last update: November 7, 2025
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The following material is not covered in the first midterm (but will be covered in the second midterm):
- Exercise 1 d), e), and f)
- Exercise 2
- Exercise 4
- Exercise 1 d), e), and f)
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The following material is not covered in the first midterm (but will be covered in the second midterm):
- Exercise 1 d), and e)
- Exercise 3
- Exercise 1 d), and e)
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The following material is not covered in the first midterm (but will be covered in the second midterm):
- Exercise 1 f)
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 1 f)
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These are some extra problems for your reference. We are sharing them as is.
- In-person in the room DIA 004 on Wed 1-4 PM and ELG 120 on Thu 9-12 AM.
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Wed (lecture): Sigma-fields and random variables (chapter 1); probability measures (section 2.1)
Thu (lecture): Probability measures and distributions (section 2.1-2.5)Corresponding Videos
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Solution sets will be available a week after problem sets are published. It is important to try doing the problems without first looking at the solution sets!
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Wed (lecture): Cantor set and the devil's staircase (section 2.5); independence (section 3.1-3.5)
Thu (exercises): Problem sets 1 and 2;Corresponding Videos
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3b1b video on music and measure theory
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Wed (lecture): expectation (chapter 4)
Thu (exercises): Problem sets 3 and 4
Corresponding Videos
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Wed (lecture): Probability couplings (chapter 5)
Thu (exercises): Review problem sets 1-4, problem set 5No Corresponding Videos - see lecture notes
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Wed (lecture): Midterm exam #1
Thu (exercises): Review of midterm exam #1
Midterm Exam #1 Statistics: (over 50)
Mean: 31.34
Median: 32
Min / Max: 8 / 48.5-
Last name A-N
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Last name L-Z
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Wed (lecture): Probability couplings (chapter 5); inequalities (chapter 6);
Thu (lecture): Inequalities (chapter 6); transform methods (chapter 7);
Corresponding Videos
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Wed (lecture): Moments; random vectors , Gaussian random vectors (sections 8.1-8.3);
Thu (exercises): Work on problem sets 5, 6, and 7
Corresponding Videos
- moments and moment generating function (no video available: see lecture notes and problem set 7, exercise 2)
- random vectors; Gaussian random vectors; joint distribution of Gaussian random vectors
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Wed (lecture): Gaussian random vectors (section 8.3); laws of large numbers (sections 9.1-9.4);
Thu (lecture): Laws of large numbers - weak and strong, proof (sections 9.4-9.6); Kolmogorov's 0-1 law
Corresponding Videos
- convergences of random variables; almost sure convergence vs convergence in probability
- Borrel-Contalli lemma; laws of large numbers; addendum
- Kolmogorov’s 0-1 law
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Solution Set 8 File
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Solution Set 9 File
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Wed (lecture): Convergence in distribution, CLT (section 10.1, 1.3 - 10.5); Application: coupon collector problem;
Thu (exercises): Work on problem sets 8 and 9
Corresponding Videos
- Convergence in distribution; equivalent definition of convergence in distribution
- The Central Limit Theorem; proof of CLT; alternative proof of the CLT;
- Application: coupon collector problem 1; and 2; St-Petersburg paradox
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Wed (lecture): Conditional expectation (chapter 11)
Thu (exercises): Work on problem sets 10 and 11
Corresponding Videos
- Conditional expectation, properties, more properties
- See also from 2023: conditional expectation and properties,
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Wed (lecture): Midterm exam #2
Thu (lecture): Martingales
