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Master 2019/2020

## Probability Theory and Mathematical Statistics

Category 'Best Course for Career Development'
Type: Bridging course (Data Science)
Area of studies: Applied Mathematics and Informatics
When: 1 year, 1 module
Mode of studies: offline
Master’s programme: Data Science
Language: English
ECTS credits: 3

### Course Syllabus

#### Abstract

The overwhelming majority of courses on data-mining and artificial intelligence implies a strong background in probability and statistics. The goal of this course is to provide those students who are not so keen on the subject matter with its fundamentals. #### Learning Objectives

• 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. #### Expected Learning Outcomes

• 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. #### Course Contents

• 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. #### Assessment Elements

• Homework 1
• Homework 2
• Homework 3
• Homework 4
• Homework 5
• Homework 6
• Exam #### Interim Assessment

• 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 #### Recommended Core Bibliography

• Dekking F. M. et al. A Modern Introduction to Probability and Statistics: Understanding why and how. – Springer Science & Business Media, 2005. – 488 pp.