MTH-202.4 Probability Theory

Probability spaces, probabilistic language: event, sample point, expectation, variance, moments, etc., random variables and their distributions, important examples: binomial, Poisson, hypergeometric, Gaussian, Cauchy, exponential, gamma and beta distributions.

Independence of events, of classes of events, of random variables. Kolmogorov's 0-1 Law, the Borel-Cantelli lemma, elementary conditional probability, Bayes formula.

Simple examples of Markov chains, Markov inequality, the weak and strong laws of large numbers.

Moment generating functions and characteristic functions, uniqueness theorem.

Inversion theorem, application to the dependence of random variables to the existence of moments.

Convergence of probability measures, tightness.

Convergence in distribution, Levy's continuity theorem, the central limit theorem for independent identically distributed summands.

Conditional expectation and conditional probability, basic theorems, regular conditional probability.

ℝ2 valued Gaussian random variables. Markov chains with countable state space: examples, transience and recurrence, stationary distributions, continuous parameter Markov chains, Poisson process, Martingales with discrete parameter, inequalities, convergence theorems, optional stopping theorem, applications in particular Markov chains.