Probability Theory and Mathematical Statistics

2024/2025

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered at:

School of Data Analysis and Artificial Intelligence

Course type:

Elective course

When:

1 year, 1 module

Instructor

Fedyanin, Denis

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

• Solve simple problems using linear and normal linear regression.
• Understand basic statistical models.
• Understand Empirical distribution function, method of moments, maximum likelihood principle.
• Understand fundamental concepts chi-squared test.
• Understand fundamental concepts of statistical hypothesis for the normal distribution.
• Understand fundamental concepts of statistical hypothesis.
• Understand fundamental concepts, advantages and limitations of Linear regression and Normal linear regression.
• Understand fundamental concepts, advantages and limitations of non-parametric statistics.
• Understand problems of statistics.

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

• Seminar 1
• Seminar 2
• Seminar 3
• Seminar 4
• Seminar 5
• Seminar 6
• Seminar 7
• Seminar 8
• Exam

Interim Assessment

• 2024/2025 1st module
  0.2 * Exam + 0.1 * Seminar 1 + 0.1 * Seminar 2 + 0.1 * Seminar 3 + 0.1 * Seminar 4 + 0.1 * Seminar 5 + 0.1 * Seminar 6 + 0.1 * Seminar 7 + 0.1 * Seminar 8

Bibliography

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.

Recommended Additional Bibliography

• Wasserman, L. All of nonparametric statistics. – Springer Science & Business Media, 2006. – 270 pp.

Authors

• Антропова Лариса Ивановна
• GROMOV Vasilii ALEKSANDROVICH
• Fedianin DENIS NIKOLAEVICH