Stochastic Simulations

elective courses

Probability theory I 1000-114aRP1a
Probability theory II 1000-115aRP2a

The course concerns computer simulation of random variables and simple stochastic processes. It comprises also an introduction to Monte Carlo (MC) methods, also known as randomized algorithms.

The subject of the course is computer simulation of random phenomena and an introduction to Monte Carlo (MC) methods, also known as randomized algorithms. The first

part of the course is devoted to methods of generating random variables with a given probability distribution and simple stochastic processes. The second part presents general ideas of designing Monte Carlo algorithms, estimating their accuracy and reducing errors. Some examples of MC algorithms actually used in scientific/statistical computations will be explained and analysed.

S. Asmussen, P.W. Glynn: Stochastic Simulation, Springer 2007

Robert, Christian, Casella, George: Monte Carlo Statistical Methods, Springer 2004

Knowledge and skills:

1. Knows the basic methods of generating random variables with various distributions: the method of transformations, rejection, composition.

2. Can generate random samples from simple probability distributions (uniform, exponential, normal, Poisson, Bernoulli) using standard functions available in a selected statistical package.

3. He can generate multidimensional random variables using the method of conditional distributions and the method of transformations.

4. Can simulate simple stochastic processes (Markov chains, Poisson processes, Markov processes on a finite state space, autoregressive processes and moving average).

5. Can calculate integrals using Monte Carlo method. He/she knows the algorithms of essential sampling, control and antithetical variables. He/she can estimate the error in Monte Carlo calculations using a consistent estimation of the asymptotic variance.

6. Knows the basic Monte Carlo methods based on Markov chains: the Metropolis-Hastings algorithm and the Gibbs probe. He/she can implement these algorithms in simple Bayesian statistical models.

Social competence:

1. Can use stochastic simulations as a tool for researching random phenomena.

2. Can present the results of the probability theory as facts about random phenomena.