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:
Exercise Sessions:
- In-person in the room INM 10 on Thu 10-12 PM.
Grading Scheme:- Graded homework - 20%
- Midterm - 20%
- Final exam - 60%
Principle for the graded homework: 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 5 pm in the dropbox outside INR 131.
Midterm Exam: Thursday, October 31, 9:15 - 11am, BC 01.
- Allowed material: one cheat sheet (i.e., two single-sided A4 handwritten pages).
Final Exam: Thursday, January 23, 9:15-12:15, INM 202
- Allowed material: two cheat sheets (i.e., four single-sided A4 handwritten pages).
- Please note that the exam content will focus more on the second part of the course (but also on the first part).
Prof. Yanina Shkel || INR 131 || yanina.shkel@epfl.ch
Teaching Assistants:Anuj Yadav || INR 034 || anuj.yadav@epfl.chCourse Webpage:References:- 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: September 10, 2024
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The solutions will be available after December 5.
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RESULTS
Max: 90/100
Mean: 61/100
Min: 34/100
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Wed: Sigma-fields and random variables (chapter 1); probability measures (section 2.1)
Thu: Probability measures (section 2.1)Corresponding Videos
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Submit the stared (*) problem in homework 1 by 5pm on Wednesday, September 18.
You can submit a hardcopy during lecture or in the drop box outside the instructor's office.
You may also use the dropbox on Moodle if you need to submit electronically.
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Wed: Probability measures and distributions (sections 2.1-2.5)
Thu: Cantor set and the devil's staircase (section 2.5); independence (section 3.1-3.3)Corresponding Videos
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3b1b video on music and measure theory
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Wed: Independence (section 3.4-3.6); expectation (chapter 4)
Thu: Expectation (chapter 4)
Corresponding Videos
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Wed: Expectation (chapter 4), characteristic function (chapter 5.1);
Thu: Random vectors (sections 6.1, 6.2)Corresponding Videos
Not covered this week: chapter 5.2
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Wed: Gaussian random vectors (sections 6.1-6.3); inequalities (section 7)
Thu: Inequalities (section 7)
Corresponding Videos
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Wed: Convergences of sequences of random variable (sections 8.1-8.4); <- covered on the midterm
laws of large numbers - weak and strong, proof (sections 8.5); <- not covered on the midtermThu: Exercise session as usual, no lecture. No graded problem set this week but we gave you some ungraded problems and practice midterms to work on.
Happy fall break!
Corresponding Videos
- convergences of random variables; almost sure convergence vs convergence in probability;
- Borrel-Contalli lemma; laws of large numbers
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This problem set is not graded.
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Wed: NO CLASS - Office Hour: 2:15pm - 3:15pm INR 219
Thu, 9:15-11:00 AM, in room BC 01: Midterm
- content: lecture notes up to (and including) section 8.4+ exercises until week 6
- allowed material: one cheat sheet (i.e., two single-sided A4 handwritten pages)-
Results
Max 54/58
Mean (without outliers) 41/58
Mean 39/58
Min 19/58
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Wed: Laws of large numbers, convergence of the empirical distribution, Kolmogorov’s 0-1 law, St-Petersburg paradox (sections 8.5-,8.8)
Thu: Convergence in distribution (section 9.1)Corresponding Videos
- Laws of large numbers; convergence of the empirical distribution, addendum; Kolmogorov’s 0-1 law; St-Petersburg paradox
- Convergence in distribution
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Wed: Convergence in distribution; equivalent definitions of convergence in distribution, Lindeberg's principle (sections 9.1, 9.3, 9.4)
Thu: No lecture, exercise session as normalCorresponding Videos
- Convergence in distribution; equivalent definition of convergence in distribution
- The Central Limit Theorem; proof of CLT
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Wed: Proofs of the central limit theorem (section 9.4, 9.5); alternative proof of CLT; application: Curie-Weiss model;
Thu: Application: coupon collector problem;Corresponding Videos
- The Central Limit Theorem; proof of CLT; alternative proof of the CLT;
- Application: the Curie-Weiss model;
- Application: coupon collector problem 1; and 2
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Wed: Hoeffding's inequality and large deviations principle (sections 10.1, 10.2); conditional expectation (section 11)
Thu: No lecture, exercise session as normalCorresponding Videos
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Wed: Conditional expectation (chapter 11), Martingales, Stopping times, Doob's optional stopping theorem (sections 12.1-12.3)Thu: Doob's optional stopping theorem (sections 12.1-12.3)
Corresponding Videos
- pre-recorded video 11.2, pre-recorded video 11.3 and 11.3b
- Martingales, stopping times, optional stopping theorem
- See also from 2023: conditional expectation and properties, optional stopping theorem
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Wed: Doob's optional stopping theorem, the reflection principle, Martingale transform (sections 12.3-12.5)Thu: Martingale convergence theorem v1 (sections 13.1-13.3)Not covered: section 12.6
Corresponding Videos (from 2023)
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Wed: MCT v1 consequences and proof (section 13.3-13.4), MCT v2 (section 13.5), Azuma's and McDiarmid's inequalities (section 13.7)Thu: Azuma's and McDiarmid's inequalities (section 13.7)Not covered: Section 13.6
Corresponding Videos (from 2023)
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Problem Set Due January 6th, 2025
