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Welcome to the webpage of the class Probability Theory at HSE, 2025/26.

Syllabus

  1. Probability spaces. Examples. Properties of probability. Continuity of probability.
  2. Conditional probability. Law of total probability. Bayes’ theorem.
  3. Independence of events: pairwise and in total. Probability model for independent trials.
  4. Borel-Cantelli lemmas.
  5. Random variables and their distributions. Types of distributions: discrete, absolutely continuous, and continuous singular. Decomposition of a distribution into discrete, continuous singular, and absolutely continuous components (without proof). Probability density. Density of a vector component given the joint density.
  6. Distribution functions and their properties. Distribution function of a discrete random variable. Relationship between the distribution function and the density of an absolutely continuous random variable. Distribution function of a random vector.
  7. Independence of random variables. Joint distribution, density, and distribution function of independent random variables. Density of the sum of independent absolutely continuous random variables. Construction of independent random variables.
  8. Mathematical expectation of a random variable and its properties. Representation of expectation as an integral over the distribution. Special cases: discrete and absolutely continuous random variables. Expectation of the product of independent random variables.
  9. Other characteristics of random variables and their properties: variance, higher moments, covariance, correlation coefficient. Median, quantiles.
  10. Generating function of a non-negative integer-valued random variable. Expressing mathematical expectation and variance through the generating function. Generating function of the sum of independent random variables. The case where the number of terms is random.
  11. The most important discrete distributions: Bernoulli; binomial, its connection to Bernoulli; Poisson, Poisson’s theorem; geometric.
  12. The most important continuous distributions: uniform, Gaussian, exponential, Cauchy.
  13. Inequalities: Cauchy-Bunyakovsky-Schwarz, Jensen’s, Markov’s, Chebyshev’s, Hoeffding’s (i.i.d. case, without proof).
  14. The Law of Large Numbers in Chebyshev’s form. The Monte Carlo method. Probabilistic proof of the Weierstrass approximation theorem for continuous functions by polynomials.
  15. Types of convergence of random variables and their relationships: almost surely, in L^p, in probability, in distribution. Proofs are required only for those implications that were proven in the lecture notes. The proof that, after a suitable “reconstruction” of the random variables, convergence in distribution implies convergence almost surely is required only in the case where the distribution functions are continuous and strictly monotonic.
  16. Weak convergence of measures and convergence of random variables in distribution. Portmanteau theorem (without proof). Prohorov’s theorem (without proof). Relationship between convergence in distribution and convergence of distribution functions.
  17. Characteristic functions and their properties. Inversion theorem. Characteristic function of a vector. Relationship with independence.
  18. Relationship between convergence in distribution and characteristic functions.
  19. Khinchin’s Law of Large Numbers and the Central Limit Theorem for independent identically distributed random variables. The Strong Law of Large Numbers: formulation.
  20. Multivariate normal distribution. Three definitions and their equivalence. Density in the non-degenerate case. Relationship between uncorrelatedness and independence for components of multivariate Gaussian vectors.
  21. Conditional expectations and probabilities with respect to sigma-algebras. Existence theorem. Conditional expectations and probabilities with respect to a sigma-algebra generated by a random variable. Conditional probability density \rho_{\xi|\eta} in the case where the random vector (\xi,\eta) is absolutely continuous.

Contents

On this website you will find

  • References for the class, including the lecture notes.
  • Exercises, including the sheets from problem-solving classes, the controls and exams sheets, and more.
  • Grading scores from the intermediate controls, and the final exams (updated during the class).
  • Additional topics, for the interested students. This includes downloadable interactive content.

Troubleshooting

If you are having troubles, you can download all the content in pdf format, although interactive content will not be available in this case.