MS-E1600 - Probability Theory D, 11.01.2021-22.02.2021

Topic outline

• Lectures and quizzes
  The course has 12 lectures, and each lecture corresponds to one chapter in the lecture notes:

  • K Kytölä, Probability Theory, 2020

  Each lecture is preceded by an online quiz which helps you to identify preliminaries from earlier courses and lectures that you may need to review independently before a lecture. Points from quizzes amount to up to 10% of the course grade, see the course syllabus for details. The deadline for completing a quiz is the start of the corresponding lecture.

  • Select activity 0 PreliminariesBefore starting the course, you are...

    0 Preliminaries

    Before starting the course, you are assumed to be familiar with basic set-theoretic notions (unions, intersections, products) and basic properties of real number sequences (limit, supremum, infimum). You are recommended to browse the following appendices where these concepts are summarized before the first lecture.
    

    • [Kytölä, Appendix A]
    • [Kytölä, Appendix B]
  • Select activity Quiz 1

    

    Quiz 1

    DL Mon 11 Jan 2021 at 10:00

  • Select activity Lecture 1: Structure of event spaces

    

    Lecture 1: Structure of event spaces Page

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    Introduction, sigma-algebra, generating a sigma-algebra, Borel sigma-algebras.

  • Select activity Quiz 2

    

    Quiz 2
  • Select activity Lecture 2: Measures and probability measures

    

    Lecture 2: Measures and probability measures Page

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    Measures and probability measures.

  • Select activity Quiz 3

    

    Quiz 3
  • Select activity Lecture 3: Random variables and measurable functions

    

    Lecture 3: Random variables and measurable functions Page

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    Measurable functions and random variables, verifying measurability, indicator functions and Dirac measures, pointwise operations on measurable real-valued functions

  • Select activity Quiz 4

    

    Quiz 4
  • Select activity Lecture 4: Information generated by random variables

    

    Lecture 4: Information generated by random variables Page

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    Function-generated sigma-algebras, Borel preimages, Doob's representation theorem, approximating by finite-range functions

  • Select activity Quiz 5

    

    Quiz 5
  • Select activity Lecture 5: Stochastic dependence and independence

    

    Lecture 5: Stochastic dependence and independence Page

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    Independent sigma-algebras, random variables, and events. Verifying independence. Borel-Cantelli lemmas.

    
    

  • Select activity Quiz 6

    

    Quiz 6
  • Select activity Lecture 6: Events of the infinite horizon

    

    Lecture 6: Events of the infinite horizon Page

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    Tail sigma-algebra, Kolmogorov's 0-1 law
    
    

    
    

  • Select activity Extended real line

    

    Extended real line File PDF

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    Details about defining the natural order, topology, and Borel sigma-algebra on the extended real line.

    
    

  • Select activity Quiz 7

    

    Quiz 7
  • Select activity Lecture 7: Integration theory

    

    Lecture 7: Integration theory Page

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    Theory of integration à la Lebesgue: definitions, monotonicity, linearity, approximation by sums
    
    

    
    

  • Select activity Quiz 8

    

    Quiz 8
  • Select activity Lecture 8: Integrations and expectations

    

    Lecture 8: Integrations and expectations Page

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    Lebesgue integrals against probability measures are expectations of random variables.  Lebesgue integrals against Lebesgue measures extend Riemann integrals.
  • Select activity Quiz 9

    

    Quiz 9
  • Select activity Lecture 9: Product spaces

    

    Lecture 9: Product spaces Page

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    Products of sets, sigma-algebras, and measures.
    
    
  • Select activity Quiz 10

    

    Quiz 10
  • Select activity Lecture 10: Random vectors and joint laws

    

    Lecture 10: Random vectors and joint laws Page

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    Random vectors, joint distributions, marginal distributions.
    
    
  • Select activity Quiz 11

    

    Quiz 11
  • Select activity Lecture 11: Convergence of random sequences

    

    Lecture 11: Convergence of random sequences Page

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    Convergence almost surely and in probability. Weak and strong laws of large numbers.
    
    
  • Select activity Quiz 12

    

    Quiz 12
  • Select activity Lecture 12: Convergence of probability measures

    

    Lecture 12: Convergence of probability measures Page

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    Fourier transforms and weak convergence of probability measures. Central limit theorem.