Characteristic functions of one dimensional random variables with applications.
Two dimensional random variables and their distributions. Marginal distributions, independence of random variables.
Moments of two dimensional random variables, covariance, correlation, a covariance matrix.
Distributions of two dimensional random variables.
Conditional distributions and linear regression.
Vector random variables, a covariance matrix.
Characteristic functions of vector random variables.
A definition of a stochastic process, finite-dimensional distributions. Kolmogorov's consistency conditions and existence of a stochastic process. Moment functions. Selected types of stochastic processes including: Markov processes, Gaussian processes, stationary and covariance stationary processes, processes with independent increments, stationary increments, orthogonal increments.Filtration,stopping time.
Markov chains: a definition and examples. Transition probabilities. Classification of states. Stationary distribution. Ergodicity.
Poisson process: a definition with basic properties, independent stationary increments, with a strong Markov property. Moment functions.
Selected processes based on the Poisson process, compound Poisson process, renewal process, non-homogeneous Poisson process.
Markov processes with continuous time and discrete state space: definition, Chapman-Kolmogorov equations, Kolmogorov's differential equations.
Wiener process: a definition with basic properties, Gaussian process, with strong Markov property. Moment functions. Properties of realizations. Orstein-Uhlenbeck process.
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