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# Probability Theory and Mathematical Statistics

2018/2019
Учебный год
ENG
Обучение ведется на английском языке
2
Кредиты
Статус:
Курс по выбору
Когда читается:
2-й курс, 1 модуль

### Course Syllabus

#### Abstract

This course is designed as an introduction to basic concepts of Probability Theory and Statistics with the emphasis on practical problems. It’s divided into two main parts Probability Theory (Module 3, year 1) and Statistics (Module 4, year 1 and Module 1, year 2). Topics include -combinatorics, -conditional probability, -random variables, -limit laws, -statistical point estimation, -hypothesis testing. The main topics are also illustrated and studied in computer statistical programs such as R, Excel, Mathematica

#### Learning Objectives

• Studying the theoretical foundations of probability theory and mathematical statistics, their practical development through the construction of mathematical models and solving statistical problems
• Understanding the types of practical problems, including those arising in sociology, which can be solved using statistical methods, and the ability to use the knowledge gained to solve them
• Ability to work with programs for mathematical calculations
• Deepening and expanding the range of knowledge about applied mathematical methods
• Mastering modern methods of data analysis, for example, basic skills of data research using statistical packages such as R (S-plus)

#### Expected Learning Outcomes

• Is able to test statistical hypothesis
• Classifies type 1 and type 2 errors
• Is able to use one-sample Z-test and one-sample and two-sample T-test
• Can test hypotheses for a mean of normally distributed data
• Can test the goodness-of-fit hypothesis using Kolmogorov test
• Can test the goodness-of-fit hypothesis using Chi-square Pearson test
• Can test homogeneity hypothesis using Two-sample Kolmogorov test
• Can test homogeneity hypothesis using Mann-Whitney rank test
• Can test Homogeneity hypothesis using Wilcoxon test for paired data
• Is able to use Chi-square test for independence
• Is able to use Fisher exact test
• Is able to use permutation tests for homodeneity and independence
• Can estimate confidence intervals by bootstrap and Jack-Knife method
• Can estimate expectations and integrals by Monte-Carlo methods

#### Course Contents

• Introduction to hypothesis testing.
• Statistical hypothesis • Statistical test • Type 1 and Type 2 errors • Significance level. Critical area • P-value
• Tests for means of normal distribution.
• One-sample Z-test • One-sample T-test • Two-sample T-test (A/B testing, model validation problems) • Double blind experiment • One-sided alternatives • Connection with the confidence intervals
• Goodness-of-fit hypothesis.
• Kolmogorov test (one sample Kolmogorov-Smirnov test) • Chi-square Pearson test • Categorical data
• Homogeneity hypothesis.
• Two-sample Kolmogorov test • Model validation problem • Mann-Whitney rank test • Wilcoxon test for paired data
• Tests for independence.
• Chi-square for independence • Fisher exact test • Lady testing tea problem
• Elements of modern statistics.
Elements of modern statistics: Permutation tests • Permutation test for homogeneity • Permutation test for independence. Again Fisher exact test Elements of modern statistics: Bootstrap, Jack-knife methods. Monte Carlo methods. • Estimation of confidence intervals by bootstrap and Jack-Knife method • Estimation of expectations and integrals by Monte-Carlo methods. Rate of convergence.

#### Assessment Elements

• Final Exam
• Activity
• Test
• Class Assignments

#### Interim Assessment

• Interim assessment (1 module)
0.088 * Activity + 0.087 * Class Assignments + 0.15 * Final Exam + 0.5 * First Year's Grade + 0.175 * Test