Probability Theory

Published

18 October 2026

The probability theory course is a direct continuation of ETMT built on the foundation of measure theory. The first part covers the language of probability theory, random vectors, convolution and covariance, types of convergence of random variables, the Borel-Cantelli lemma, the zero-one law, and laws of large numbers. The second part covers weak convergence of measures, characteristic functions, the central limit theorem, Gaussian vectors, and conditional expectation together with conditional distributions.

Prerequisites

The course builds on β€œFoundations of Probability and Measure Theory”: probability space, random variables, the Lebesgue integral.

Assessment format

Assessment follows the current rules of the semester (tests, colloquium, exam); see the department’s current schedule.

Contents

Part 1

  1. The language of probability theory
  2. Random vectors
  3. Convolution, covariance
  4. Convergence of random variables
  5. The Borel-Cantelli lemma, zero-one laws
  6. The law of large numbers

Part 2

  1. Weak convergence
  2. Characteristic functions
  3. The central limit theorem
  4. Gaussian vectors
  5. Conditional expectation
  6. Conditional distributions

See also the course program.

About this text

This text is based on notes from seminar sessions.

Author: M.A. Nedoshivin
Email: nedoshivin.ma@phystech.edu

Found a typo or an error? Email the address above.

The materials are distributed under the CC BY-NC-SA 4.0 license; see the β€œAbout the project” page for details.