Теорія ймовірностей та математична статистика (Eng.)

Overview: Probability and statistics have become the basis for data science with its

numerous applied techniques. Our aim within this course is to introduce the

students to the main concepts and methods of probability and statistics and to

help them develop probabilistic and statistical thinking. We will discuss the basic

notions of probability (sample space and events, axioms of probability,

independence and conditioning; discrete and continuous random variables and

their distributions; expectations, variance and other characteristics) and statistics

(samples, descriptive statistics, parameter estimation, hypothesis testing and

regression) that are necessary for understanding the main techniques of data

science.

Course topics:

Part I. Basic notions

Topic 1. Introduction

● Notion of probability

● Classical and geometric probability

● Combinatorial analysis

Topic 2. Axioms of probability

● Sample spaces, events, probability

● Axioms of probability

● Inclusion-exclusion principle

Topic 3. Conditioning

● Conditional probability; Independence

● Total probability rule

● Bayes’ formula

Part II. Random variables

Topic 4. Discrete random variables

● Definition and examples

● Probability mass function and cumulative distribution function

● Independence and joint distribution

Topic 5. Continuous random variables

● Density and cumulative distribution functions

● Standard continuous distributions

● Independence, joint distributions, transformed distributions

Topic 6. Basic characteristics of random variables

● Expectation and variance; moments

● Covariance and correlation

● Conditional distributions

Part III. Limit theorems and random processes

Topic 7. Limit theorems: LLN and CLT

● Chebyshev’s inequality and Weak Law of Large Numbers

● Types of convergence; SLLN; Monte Carlo method

● Central Limit Theorem and approximation by normal distribution

Topic 8. Markov chains

● Examples and basic notions

● Stationary distribution

● Absorption probability and time

Topic 9. Some random processes

● Bernoulli process

● Poisson process

● Random walk

Part IV. Parameter estimation

Topic 10. Statistical models

● Statistical models

● Samples and their characteristics; graphical tools

● Parameter distribution families, statistics, and estimators

Topic 11. Parameter estimation

● Unbiased and consistent estimators

● Moment method

● Maximum likelihood estimator

Topic 12. Confidence intervals

● Point vs interval estimation

● Confidence intervals for the mean and variance

● Exit polls and confidence intervals

Part V. Hypothesis testing

Topic 13. Hypothesis testing

● Neyman-Pearson framework

● Errors of types I and II

● Test size, power function, p-value

Topic 14. Tests for normal distribution

● The z-test

● The t-test

● Tests for the variance

Topic 15. Regression

● The linear model

● Parameter estimation

● Hypothesis testing in linear regression