# Probability Theory (CS program)

## Description

### Overview

This is a page of my course for the third-year students of Bachelor’s Program “Computer Science” at MIPT.

This is the second part of the course. The first part was devoted 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 were introduced and we proved the Tchebyshev's form of the weak law of large numbers.

This 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.

## Homeworks

Homework 1. Discrete probability.

Homework 2. Conditional probability.

Homework 3. Measure theory. Total probability formula.

Homework 4. Bernoulli scheme. Independence of events.

Homework 5. Probability distribution. CDF

Homework 6. Probability density function.

Homework 7. Random vectors

Homework 8. Expected value

Homework 9. Characteristic functions

Homework 10. Central Limit Theorem

## Attendance & Marks

Attendance list.

Marks.

### Prerequisities

Probability Theory (part 1), Basic set theory, basic measure theory, calculus, mathematical analysis, combinatorics.

### Program

This is a preliminary version of the program. Some small changes are possible during the semester.

- Definition of an algebra of sets and a sigma-algebra. Definition of a Measurable space. Additive measure. Lebesgue measure. Probability space.
- General definition of a random variable.
- Cumulative distribution function and its properties.
- General definition of Expected value of a random variable. Independence of random variables in general case.
- Absolutely continuous random variables. Probability density function. Examples of continuous probability distributions: Uniform distribution, Exponential distribution, Normal distribution, Cauchy distribution.
- Joint probability distribution for two continuous random variables. Joint probability density distribution for two continuous random variables
- Characteristic functions and its properties. Characteristic function of a sum of independent random variables. Inversion formula.
- Convergence in distribution. Lévy's continuity theorem. Weak law of large numbers ( Khinchin's form).
- Central limit theorem.

## Lectures

**Lecture 1** [03.09.2021]

**Lecture 2** [03.09.2021]

**Lecture 3** [17.09.2021]

**Lecture 4** [24.09.2021]

**Lecture 5** [01.10.2021]

**Lecture 6** [08.10.2021]

**Lecture 7** [15.10.2021]

**Lecture 8** [22.10.2021]

**Lecture 9** [29.10.2021]

**Lecture 10** [05.11.2021]

**Lecture 11** [23.11.2021]

**Lecture 12** [26.11.2021]

**Lecture 13** [03.12.2021]

**Lecture 14** [10.12.2021]

## Course guidelines and grading system

At the end of this course, you will get a grade from 1 to 10 (you need at least 3 to pass) according to the following parameters:

- Max of 1 point for class attendence (1 if you missed no more than 2. And o,5 if you missed no more than 3)
- Max of 3 points for Test 1 [November 12]
- Max of 3 points for Test 2 [December 12]
- Max of 3 points for final exam on theory. [December 17]

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 simple problem to raise your grade to z+1. Otherwise you get just z.

Every week, after class, I will upload a new homework. Each of you have to solve (or at least try to solve) these problems. I will randomly choose one of you during to seminar so that you can explain to everyone your solution.

## References

- Probability, Shiryaev Albert N.
- Introduction To Probability, Joseph K. Blitzstein, Jessica Hwang.
- Курс теории вероятностей и математической статистики, Севастьянов Б.A.
- Курс теории вероятностей, Чистяков В.П.
- Мера и интеграл, Дьяченко М.И.

## Recommended Material

- Short lectures on measure theory: [playlist]

- Here are interective videos explaining concepts from probability: