Probability Theory and Stochastic Processes

See semester study programme.

The aim of the course is to present advanced techniques of the probability theory and selected topics in the theory of stochastic processes.The course serves as a foundation for more advanced courses on applied probabilistic methods in mathematical statistics, game theory, financial markets modeling, applications of mathematics in finances and actuarial sciences.

Literatura podstawowa:

J. Rice, Mathematical Statistics and Data Analysis, Duxbury 2006; J. Jacod, P. Protter, Probability Essentials, Springer-Verlag 2013.

Literatura uzupełniająca:

G.R.Grimmett, D.R.Stirzaker, Probability and Random Processes, Oxford University Press1992; R.Durrett, Essentials of Stochastic Processes, Springer-Verlag 1999.

Wiedza:

Students should know the definition of a characteristic function of one and multidimensional random variable.

Students should know the following concepts: the distribution of a multidimensional random variable, the cumulative distribution function, marginal distributions, moments of a multidimensional random variable, independence of random variables, conditional distribution, and conditional expectation.

Students should know the definition of a stochastic processes, finite-dimensional distributions and moment functions, a process with idependence increments,a process with stationary increments, a stationary process, a Gaussian process, a Markov Process. Students should know basic properties of a Markov chain, a Poisson process, a Markov process with discrete state space, and a Wiener process.

Umiejętności:

Students should be able to calculate a characteristic function of one dimensional random variable and to identify a distribution of a random variable by its characteristic function. Moreover students should be able to calculate moments of a random variable using its characteristic function,

Students should be able to compute a cumulative distribution function, a probability mass function,a density function of a two dimensional random variable and a similar characteristics of marginal distribution. Moreover students should be able to compute a distribution of a function of two dimensional random variable, conditional distribution, conditional expectation and to check the independence of random variables.

Students should be able to check in simple cases whether a stochastic process is a process with independent increments, with stationary increments, a stationary process, a Markov process, a Poisson process, a Wiener process. Moreover students should be able to compute the probability of some events related to a Markov process and a Poisson process.

Kompetencje społeczne:

The appretiation of systematic self-study.

The exposure on precise and logical way of thinking

egzamin tradycyjny-pisemny: 50.00%

kolokwium: 50.00%