Weekly outline

  • General

    "Probability theory is nothing but common sense reduced to calculation." Pierre-Simon de Laplace, 1812 (see other interesting quotes from Pierre-Simon de Laplace)

    (Zoom link in case of online sessions)

    SwitchTube channel     for the course (please note that these videos were made for a previous version of the course: there will be some differences with this year's version)

    Lectures:
    - In presence in room CM 1 120 on Wed 2-4 PM and in room INR 219 on Thu 8-10 AM

    Exercise sessions:
    - in presence in room INR 219 on Thu 10-12 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.

    Course instructor:
    Olivier Lévêque     // LTHI // INR 132 // 021 693 81 12 // olivier.leveque@epfl.ch

    Teaching assistants:
    Bora Dogan // LINX // bora.dogan@epfl.ch
    Anand George // SMILS // anand.george@epfl.ch
    Lifu Jin // lifu.jin@epfl.ch

    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.

  • Week 1 (February 22-23)

    Wed: Sigma-fields (sections 1.1, 1.2, 1.3)
    Thu: Random variables (sections 1.4, 1.5), probability measures (section 2.1)

  • Week 2 (March 1-2)

    Wed: Distributions of random variables (sections 2.2, 2.3, 2.4, 2.5)
    Thu: Independence (sections 3.1, 3.2, 3.3, 3.4)

  • Week 3 (March 8-9)

    Wed: Do independent random variables really exist? (section 3.5), convolution (section 3.6)
    Thu: Expectation (section 4)

  • Week 4 (March 15-16)

    Wed: Characteristic function (section 5), random vectors (section 6.1)
    Thu: Gaussian random vectors (sections 6.2, 6.3)

  • Week 5 (March 22-23)

    Wed: Inequalities (section 7)

    Thu: Convergences of sequences of random variables and Borel-Cantelli lemma (sections 8.1, 8.2, 8.3, 8.4)

  • This week

    Week 6 (March 29-30)

    Wed: Laws of large numbers - weak and strong, proof (section 8.5)

    Thu: Convergence of the empirical distribution, Kolmogorov's 0-1 law (sections 8.6, 8.7)

    NB: Because of the midterm next week, Hwk 6 is due on Thursday, April 20 only!

  • Week 7 (April 5-6)

    Wed: St Petersburg's paradox and extension of the weak law (section 8.8)
    Thu, 8:15-10:00 AM, in room CM 1 4: Midterm
    - content: all the course + exercises until week 6 (except Hwk 6)
    - allowed material: one cheat sheet (i.e., two single-sided A4 handwritten pages)

    • Week 8 (April 19-20)

      Wed: Convergence in distribution and equivalent definitions (sections 9.1, 9.3)
      Thu: Lindeberg's principle and proof of the central limit theorem (section 9.4)

      • Week 9 (April 26-27)

        Wed (online): classical proof of the central limit theorem (section 9.5, video), introduction to moments (section 10, video)
        Thu: Moments and Carleman's theorem (section 10), Hoeffding's inequality (section 11.1)

        • Week 10 (May 3-4)

          Wed (online): Hoeffing's inequality (section 11.1, video) and large deviations principle (section 11.2, video)
          Thu: Conditional expectation (section 12)

          • Week 11 (May 10-11)

            Wed (online): Martingales: definition and basic properties (section 13.1, video 1, video 2)
            Thu: Stopping times and optional stopping theorem (sections 13.2, 13.3)

            • Week 12 (May 17)

              Wed: Martingale transforms and Doob's decomposition theorem (sections 13.5, 13.6)
              Thu: Ascension (no class)

              In replacement, there will be an exercise session on Monday, May 22, 10-12 AM (room TBA).

              • Week 13 (May 24-25)

                Wed: Martingale convergence theorem (v1) and consequences (sections 14.1, 14.2, 14.3)
                Thu: Proof of the theorem (section 14.4)

                • Week 14 (May 31 - June 1)

                  Wed: MCT (v2), and generalization to sub- and supermartingales (sections 14.5, 14.6)
                  Thu: Azuma's and McDiarmid's inequalities (section 14.7)