| Goal: To lay the foundations of probability theory and statistics |
| Course description: Kolmogorov probability space; law of total probability; conditional probability; Bayes’ theorem; probability distribution function; expectation, variance and moments; special distributions (Poisson, uniform, etc.). Moment generating function, characteristic function. 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. Estimation techniques, maximum-likelihood estimation, method of moments, method of least squares. Hypothesis testing; confidence intervals. Parametric and nonparametric tests. |
