Syllabus
A summary of basic concepts of mathematical statistics: the probability measure, random variables, probability distribution functions, convergence of random variables, conditional and marginal probabilities, random samples, the expectation value, the variance, the covariance matrix, the coefficient of linear correlation, higher moments, the characteristic functions and their use, transformation of random variables, the convolution.
The Bayes theorem, the Tchebycheff inequality and the Central Limit theorem.
A survey and properties of the most relevant statistical distributions for the needs of processing the data from physical measurements. Mutual relations between some of these distributions.
Statistics for estimating parameters and their main characteristics (consistency, unbiasedness and sufficiency). Selected methods for constructing these statistics. Fisher's statistical information and the Rao-Cramér inequality.
Parameter estimation by the method of the least squares (the linear model and a sub-model, testing their validity). Gauss-Markov theorem. Properties of residuals of the fit, bands of reliability. The chi-square test. Smoothing the experimentally assessed functions with the aid of the regression polynomial. The use of Forsythe polynomials for suppression of numerical instabilities. Selected iteration methods for finding the least square in case of non-linear models.
The maximum likelihood method and its use for parameter estimation.
Simulation of random processes by the Monte Carlo Method. Deterministic generators of random numbers. Basic methods for generating random samples (the method of the inverse distribution functions, the method of von Neumann, methods generating discrete random quantities). The use of the Monte Carlo Method for a design of an experiment and evaluation of experimental data.
The Bayesian use of Bayes theorem (the prior and posterior probability densities, the procedure for estimating unknown parameters). Illustrating examples of the use of the Bayesian approach.
Annotation
Basic concepts in probability, random quantities, probability distributions. Parameter estimation: maximum likelihood and least square techniques, testing of hypotheses, Monte Carlo modelling, basic methods of data processing.