# Probability Theory (MSc Medical Biotechnology)

## Description

### Overview

This is a page of my course for the first-year students of Master’s Program “Medical Biotechnology” at MIPT.

This is the first part of the course. This semester will be 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 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.

### Prerequisities

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.

1. Elements of combinatorics. Permutations and combinations. Inclusion-Recursion.

2. Combinatorial probability. Discrete probability models. Geometrical probability model. Conditional probability. Independence of events, pairwise and mutually independent events. Total probability formula. Bayes theorem.

3. Bernoulli trials scheme. Notion of random graphs.

4. General probability. Kolmogorov axioms.

5. Random variables. Probability mass function and their properties. Discrete random variables. Basic types of discrete probability distributions. Examples of probability distributions: binomial, Poisson, geometric, hypergeometric).

6. Definition and properties of the expectation and variance of a random variable. Covariance and correlation.

7. Tchebychev and Markov inequalities. Existence of a triangle in a random graph.

8. Law of large numbers (weak version).

## Homeworks

**Homework 1.** Basic combinatorics.

**Homework 2-3. **Inclusion-exclusion formula.

**Homework 4.** Classical probability

**Homework 5. **Classical probability.

**Homework 6.** Conditional probability.

**Homework 7.** Bayes Formula

**Homework 8. **Probability distribution

**Homework 9. **Expected value. Variance.

**Homework 10.** Random graphs

Download all homeworks [here]

## Attendance & Marks

**Attendance list.**

**Marks.**

## 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: (by the way, you can convert this grade into the 1-5 scale in the following way: 1-2 (F*ail *or *2/5*), 3-4(P*ass *or* 3/5*), 5-7 (Good* *or* 4/5*), 8-10(E*xcellent *or* 5/5*) )

- 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 25]
- Max of 3 points for Test 2 [January 20]
- Max of 3 points for final exam on theory [January 27]

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.

Since our group is composed of more than 40 students, I will **check attendance** in the following way: During our meeting on Zoom, you need to write on the chat your name followed by the number you are in the list of attendance. For example, if you are third on the list, you should write *[your name] - 3.*

## 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.