Теория вероятностей и математическая статистика

2026/2027

Статус: Маго-лего

Кто читает: Департамент статистики и анализа данных

Где читается: Факультет экономических наук

Охват аудитории: для своего кампуса

Преподаватели: Жабир Жан-Франсуа Мехди, Катышев Павел Константинович

Язык: русский

Кредиты: 6

Контактные часы: 96

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Аннотация

This course aims to provide a solid introduction to probability theory and mathematical statistics. The fundamental concepts and mathematical tools for modeling and analysis of random phenomena will be presented and discussed.

Цель освоения дисциплины

To provide students with the knowledge of the theoretical aspects and modeling tools related to probability theory.
To provide students with the knowledge of elementary techniques to analyze probabilistic models.
To present and study some fundamental distributions of random variables.
To develop students' ability to apply the knowledge acquired during the course and to use probabilistic models in concrete situations, recognizing the appropriate frameworks and analytical tools related to the study.
To outline the basic concepts and methods of mathematical and applied statistics.
To give practical skills in applying statistical methods in applied research.
To give an idea of the applied methods of multidimensional statistical analysis.

Планируемые результаты обучения

The concepts of random variables will be generalized to random vectors. It will be introduced the concepts of multivariate distributions.
The students are expected to learn the concept of random variable, some distribution functions, expected value and variance. Also, they will have the opportunity to relate random problems with random variables.
The students are expected to learn the operations between sets, the notion of σ-algebra, Kolmogorov's axioms and properties of probability measures.
The students will have the opportunity to model random experiments where the knowledge of continuous random variables is required. The concepts of expected value and variance for continuous random variables will be generalized. Also, the students will learn some of the main inequalities in Probability Theory.
The students will learn about some convergences that are used in Probability Theory. LLN and CLT and their applications will be discussed.

Содержание учебной дисциплины

Introduction to Probability Theory
Discrete random variables
Continuous random variables
Random vectors
Convergences, LLN and CLT
Statistical estimation of parameters. Samples. Property of estimators. Unbiasedness, efficiency, consistency
Interval estimation. Standard confidence intervals for the parameters of a normal population. Confidence intervals for the mean, variance, difference of means, variances ratio, population proportion, the difference of proportions. The sample size
Hypotheses testing. Type I, type II errors. P-value of the test. Tests on the values of the parameters of the normal population. Tests on mean, variance, differences, difference of means, variances ratio, population
Estimation methods. The Method of Moments. Maximum Likelihood method. Their properties, examples. Information inequality (Fisher)
Test statistics, Neumann-Pearson Lemma. Likelihood ratio test. Wald test. Lagrange multiplier test
Goodness-of-fit tests. Contingency tables. Kolmogorov-Smirnov test
Bayes approach to estimation
One- and Two-way ANOVA
Some concepts of non-parametric methods. Wilcoxon tests, run test. Rank correlation coefficients
Sufficient statistic. Minimal sufficient statistic. Rao-Blackwell Theorem. Complete statistics. Lehmann-Sheffe Theorems

Элементы контроля

Seminar assessment - Theory of Probability
Mid-module 1 Test
Midterm Exam - Theory of Probability
Final examination of the part "Theory of Probability"
Written homework
Quiz
Written exam work

Промежуточная аттестация

2026/2027 2nd module
The final grade of the course is calculated as an average of the evaluations of the two parts/modules of the course with 50% for the part "Theory of Probability" (module 1) and 50% for the part "Mathematical Statistics" (module 2). The grade for the course "Theory of Probability" is computed according to the formula: 40%*N1 and 60%*N2 with N2="Mid-term Exam - Theory of Probability", and N1=max(Mid-module 1 test, 60% Mid-module 1 test+40% Seminar assessment"). The grade for the course “Mathematical Statistics” is calculated using the formula: 20% M1 + 20% M2 + 60% M3, where M1 is the grade for tests, M2 is the grade for homework, M3 is the grade for the exam paper.

Список литературы

Авторы

Катышев Павел Константинович
Жабир Жан-Франсуа Мехди

Программа дисциплины