Goal: To lay the foundations of probability theory and statistics
Course description: Kolmogorov probability space; law of total probability; Bayes’ theorem. Random variables and their properties; probability distribution function; expectation, variance and moments. Transforms of distributions (generating functions, characteristic function, Laplace-transform). Joint distributions; random vectors; independence; covariance matrix. General definition and properties of conditional expectation; law of total expectation. Types of convergence; Borel-Cantelli lemmas; laws of large numbers; sums of random variables; central limit theorems. Statistical space; sample; statistics; ordered sample; empirical distribution function; Glivenko-Cantelli theorem. Unbiased, efficient and consistent estimator; sufficiency, completeness and ancillarity; Rao-Blackwell theorem. Estimation techniques, maximum-likelihood estimation, method of moments, method of least squares. Bayesian estimation. Hypothesis testing; confidence intervals. NeymanPearson lemma; parametric and nonparametric tests.
Probability theory and the basics of mathematical statistics