Probability Theory and Mathematical Statistics
- To introduce the theoretical foundations of Probability theory.
- To introduce the theoretical foundations of Mathematical statistics.
- To provide the students with practical skills of modelling real-world in the framework of probability and statistics.
- Understand problems of statistics.
- Understand basic statistical models.
- Understand Empirical distribution function, method of moments, maximum likelihood principle.
- Understand fundamental concepts of multivariate normal distribution.
- Understand fundamental concepts of statistical hypothesis.
- Understand fundamental concepts of statistical hypothesis for the normal distribution.
- Understand fundamental concepts chi-squared test.
- Understand fundamental concepts, advantages and limitations of non-parametric statistics.
- Understand tests for power-law distributions.
- Understand fundamental concepts, advantages and limitations of Linear regression and Normal linear regression.
- Solve simple problems using linear and normal linear regression.
- Problems of Statistics. Parameter estimators. Basic statistical models. Distributions and sample statistics. Estimators. Unbiased estimators. Unbiased estimators for expectation and variance. Consistency. Efficiency and mean-squared error. Kramer-Rao inequality.
- Empirical distribution function. Empirical parameters. Method of moments. Maximum likelihood principle. Likelihood and loglikelihood.
- Multivariate normal distribution. Student and Fisher distributions. Normal sample.
- Testing statistical hypothesis. Null hypothesis. Test statistics. Type I and type II errors. Significant level. Critical region and critical values.
- Statistical hypothesis for the normal distribution. The one-sample t-test. Comparing two samples. Tests for variance.
- Chi-squared test. For known parameters. For unknown parameters.
- Non-parametric statistics. Sign test. Two-sided sign test. Wilcoxon sign-rank test.
- Tests for power-law distributions. Hill test. Ibragimov’s corrections for confident intervals. Data-collapse approach. Clauset-Shalizi-Newman test.
- Linear regression. Normal linear regression.
- Interim assessment (1 module)0.4 * Exam + 0.1 * Homework 1 + 0.1 * Homework 2 + 0.1 * Homework 3 + 0.1 * Homework 4 + 0.1 * Homework 5 + 0.1 * Homework 6
- Dekking F. M. et al. A Modern Introduction to Probability and Statistics: Understanding why and how. – Springer Science & Business Media, 2005. – 488 pp.
- Wasserman, L. All of nonparametric statistics. – Springer Science & Business Media, 2006. – 270 pp.