The notion of probability, elementary properties. Kolmogorov probability field Combinatorial calculation of probabilities. Conditional probability, properties, calculation. Bayes theorem. Independency. Random (vector)variable and its distribution, joint distributions. Independent random variables. Random walk and ruin probabilities. Particular discrete distributions. Mean and variance, properties, calculation, inequalities. Median, moments. Covariance and the coefficient of correlation. Distribution and density functions. The distribution of sums of independent random variables (convolution). Particular absolute continuous distributions and their properties. Weak law of large numbers. Central limit theorem. Normal and multivariate normal distribution.
Statistical space, samples, statistics. Ordered statistics, empirical distribution functions. Unbiased, efficient and consistent estimators. Complete and sufficient statistics. Neyman factorization theorem. Fisher information, Cramer-Rao inequality. Rao-Blackwell-Kolmogorov theorem. Confidence intervals. Maximum likelihood estimators, properties. The method of moments. Hypothesis testing. Comparison of tests. Randomized and sequential tests. The Neyman-Pearson lemma. U-, Student t-, and F-tests. c2 -test, and its applications. Linear regression, and the method of least squares. The simplest cases of variance analysis.