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# Probability Theory and Mathematical Statistics

2020/2021
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты
Статус:
Курс обязательный
Когда читается:
1-й курс, 3, 4 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) and Statistics (Module 4). 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

• 1. Studying the theoretical foundations of probability theory and mathematical statistics, their practical development through the construction of mathematical models and solving statistical problems 2. 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 3. Ability to work with programs for mathematical calculations 4. Deepening and expanding the range of knowledge about applied mathematical methods 5. Mastering modern methods of data analysis, for example, basic skills of data research using statistical packages such as R (S-plus)

#### Expected Learning Outcomes

• describes a random experiment and a sample set
• performs computations of conditional probabilities
• constructs mathematical model, describing a given random experiment
• applies tables of standard normal distributions to compute probabilities and quantiles
• correctly constructs the null and the alternative hypothesis
• calculates sample mean, sample variances, sample proportion, sample quantiles and is able to do this by means of computer programs
• constructs confidence intervals for the means and proportions and explains their meaning

#### Course Contents

• Combinatorics and Bernoulli trials.
History of Probability, Probability and Data Analysis, Random experiment. Outcomes. Events, Operations with Events, Statistical definition of probability, Axiomatic definition of probability, Classical formula of probability for equally likely outcomes, Inclusion-exclusion formula, Birthday paradox, Independent and dependent events, Bernoulli trials, Formula for the number of successes, Erdos-Renyi random graphs
• Conditional probability. Bayes' formula.
Definition of conditional probability. Illustration via contingency tables, Multiplication formula for probabilities, Chain rule, Independence via conditional probability, Law of Total Probability, Monty Hall Paradox, Bayes’ Theorem, Prior and Posterior probabilities
• Discrete and continuous random variables.
Discrete random variables, Probability mass function, Binomial random variable. Newton Binomial formula, Hypergeometric random variable, Poisson random variable. Rare events. Poisson limit theorem, Expectation. Variance. Standard deviation, Mode. Median, St.Petersburg paradox, Continuous Random Variables, Cumulative distribution function, Density of continuous random variable, Uniform, Normal, and Exponential random variables, Formulas for expectation and variance, Independence and dependence of random variables. Joint distribution
• Normal distribution and Central Limit Theorem
Family of Normal distributions. Standard Normal distribution. Standardization. Z-score. Quantiles of Normal variables. Central Limit theorem. Applications in insurance, stock market, etc, Law of Large Numbers
• Statistical Point estimation.
Main definitions: population, sample, representative sample, Frequency histogram, Cumulative empirical distribution function, Probability vs Statistics, Properties of Point estimates: unbiasedness and consistency, sample quantiles, sample mean, sample variance, unbiased sample variance, sample proportion, Outliers, Correlation, Statistics in R/Python
• Interval estimation.
Point estimates vs Interval estimates, Confidence interval for the mean, One-sided confidence intervals, Confidence intervals for proportions
• Hypothesis testing.
Statistical hypothesis, Statistical test, Type 1 and Type 2 errors, Significance level. Rejection Region, P-value, Hypotheses for a mean of normally distributed data, Z-test. Connection with the CLT, Connection with the confidence intervals, Student T-distribution. Two-sample T-test, Applications: A/B testing, Model validation problems, Double blind experiment, Homogeneity hypothesis, Mann-Whitney rank test, Two-sample Kolmogorov test

#### Assessment Elements

• Test 1 (80 minutes)
• Test 2 (80 minutes)
• Small-tests (30-minutes)
• Self-Study (report)
• Final Exam – 60 min written examination
Conditions for writing Exam: In case of online exam: At the exam it is forbidden to use any notes and any copybooks, as well as any gadget with internet. The students who violate these rules will get a warning, if they violate the rules twice they will get 0. In case of on-line exam: All students must have their cameras switched on during the exam. It is strongly recommended to send solution of the tests in .pdf format.

#### Interim Assessment

• Interim assessment (4 module)
Grading Policy: Activity grade — 11%, Test №1 — 18%, Test №2 — 18%, Average of all Small-test — 18%, Self-study (report) — 10%, Final Exam — 25%.

#### Recommended Core Bibliography

• A first course in probability, Ross, S., 2010
• Essentials of statistics for the behavioral sciences, Gravetter, F. J., 2014