# Probability Theory

#### Overview

Welcome to my course on Probability Theory for the second-year students of Bachelor’s Programs “Computer Science” and “Biomedical Engineering” at MIPT.

This is the first part of the course. We will devote this semester to the study of discrete probability theory, incluiding introductory notions of random variables and the way of computing their expected value, variance and correlation coefficient. In addition, the Markov and Tchebyshev inequalities will be introduced and we will prove the Tchebyshev's form of the weak law of large numbers.

Next semester, we will generalize the concept of a random variable as a measurable function. We will extensively use the CDF and the PDF of random variables to compute all their interesting numerical characteristics. As a whole, we will completely construct the concept of a propability space. We will conclude our course with the proof of one of the main results in probability theory - the central limit theorem.

## Problem sets

*Every seminar we cover one of the following problem sets.*

Problem set 1. Combinatorics.

Problem set 2. Classical probability.

Problem set 3. Geometric probability.

Problem set 4. Bernoulli scheme and independence.

Problem set 5. Conditional probability. Bayes Formula.

Problem set 6. Discrete random variables. Joint distributions.

Test 1 [March 21]

Problem set 7. Absolutely continuous Distributions.

Problem set 8. Random variables and random vectors

Problem set 9. Independence. Convolution formula

Problem set 10. Expected value and Variance of Discrete Distributions

Problem set 11. Expected value and Variance of Continuous Distributions

Test 2 [May 8]

Exam (зачет) [May 16]

#### Topics discussed in lectures

**Lecture 1. **

## Attendance & Marks

Marks.

#### Course guidelines and grading system

At the end of this course, you will get a grade from 0 to 10 (you need at least 3 to pass) according to the following parameters:

- A= Max of 1 point for class attendance (1 if you missed no more than 1 class. And o,5 if you missed no more than 2)
- P= Max of 2 points for participation in class by solving homework on the board.
- T= Max of 6 points for Test 1 + Test 2
- E= Max of 4 points for final exam (зачет) on theory

Final grade = A+P+T+E-2

**IMPORTANT: ***You must get at least 1.5 points in the final exam on theory in order to pass the course.*

The theoretical exam and tests 1 and 2 are ** closed-book**, that is, you are not allowed to use any material.

The number of points you get for each activity, is either an integer x or x+0.5.

If your final grade (after the final exam) is not an integer (z+0,5), you can solve an extra problem to raise your grade to z+1. Otherwise you get just z.

If you have <= 2 points for A+T, then you have a solve problems in the exam. You need a minimum number of points of the solution of these problems to get your theoretical questions and continue with the exam.

On the other hand, in the retake you have to solve problems independently from the number of points you have for T.

#### Recommended literature

(Graduate Texts in Mathematics) 2nd Edition - Albert N. Shiryaev.**Probability**- Stanley H. Shan. [download]**Introduction to probability for Data Science**- Carlos Fernandez-Granda.**Probability and Statistics for Data Science**- Joseph K. Blitzstein, Jessica Hwang.**Introduction To Probability**- Мера и интеграл, Дьяченко М.И.
- Курс теории вероятностей и математической статистики, Севастьянов Б.A.
- Курс теории вероятностей, Чистяков В.П.

#### Recommended extra material

- Short lectures on measure theory: [playlist]
- Short lectures on Probability Theory [playlist]
- Probability theory course IMPA [playlist]
- Probability theory course Harvard University [playlist]
- Interactive videos on probability from 3Blue1Brown [video]
- Lectures in introduction to probability (in russian) [playlist]
- [link]