# Probability Theory II (Medical Biotechnology)

#### Overview

This is the second part of the course. 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.

## Problem sets

Problem set 1. Random walk.

Problem set 2. Probability distribution. CDF and PDF.

Problem set 3. Random variables and random vectors.

Problem set 4. Independence. Convolution Formula.

Problem set 5. Expected value and variance.

Problem set 6. Characteristic functions.

Problem set 8. Limit theorems and convergence of random variables.

## Attendance & Marks

#### Topics discussed in lectures

**Lecture 1. [02/03] **Simple symmetric random walk. P(Sn=k). P(S1>0,...Sn=k).

**Lecture 2. [09/03] **Formal definition of probability distribution. Borel sigma-algebra. CDF. Properties of CDF. Absolutely continuous distributions. PDF. Properties of PDF. Examples: Unifom, Exponential, Normal.

**Lecture 3. [16/03] **Review of lecture 2.

**Lecture 4. [23/03] **Probability distributions in Rn. CDF in Rn. Properties of CDF in Rn. Marginal distribution. Marginal CDF. Independence of distributions and CDFs. Absolutely continuous distributions in Rn. PDF of a vector with independent components. Properties of PDF.

**Lecture 5. [30/03] **Definition of random elements. Borel functions. Borel function of a random vector. Theorem of a random vector as a vector of random variables. Definition of independence of random variables. Criterion of independence. Independence of funcions of independent random variables.

**Lecture 6. [06/04] **Convolution for sum and division of two independent random variables. Examples with the uniform distributions.

**Lecture 7. [20/04] **Expectation as EX=EX^{+} -EX^{-}. Lebesgue Integral. Example of discrete random variable without expectation. Expectation of absolutely continuos random variables. Example with normal distribution.

**Lecture 8. [27/04] test 1**

**Lecture 9. [04/05] **Convergence of random variables. Convergence a.s. with example. SLLN. Convergence in probability and relation with WLLN. Example when we have convergence in probability but not a.s.. Convergence in distribution. Example with poisson limit theorem and Moivre-Laplace theorem.

**Lecture 10. [11/05]** Slutsky's theorem. Limit theorems. Law of iterated logarithm. Characteristic functions. Seminar: Finish problem set 5.

**Lecture 11. [18/05] **Properties of Characteristic funcions and main theorems. Seminar: Start problem set 6.

**Lecture 12. [25/05] [classroom 416 ГК] ***Levy's continuity theorem. CLT. Seminar: Start problem set 7. *

**Lecture 13. [01/06] **Gaussian vectors. *CLT for vectors. Seminar: Finish problem set 7.*

**June 8 - Test 2**

**June 15 - EXAM**

#### 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 attendance (1 if you missed no more than 1. And o,5 if you missed no more than 2)
- Max of 3 points for Test 1
- Max of 3 points for Test 2
- Max of 4 points for final exam on theory

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

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.

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.

#### Syllabus

- Probability space. Properties of probability measure. Probability distribution. CDF and properties. Examples (discrete). Absolutely continuous distributions. PDF. Examples. Uniform, exponential, normal.
- Distributions in Rn. CDF and Properties. Independence and marginal distributions. PDF in Rn and Properties.
- Random elements. Random vector as a vector of random variables. Independence of random vectors.
- Criteria of Independence. Fubini’s theorem. Convolution formula.
- Expected value in general. Properties. Examples. Change variable theorem. Variance and examples. Jensen inequality.
- Conditional expectation. Properties.
- Conditional distribution.
- Convergence of random variables. Connection between different types of convergence. Examples where they doesn’t hold.
- Criteria of convergence. Borel-Cantelli lemma.
- Law of the iterated logarithm. SLLN 1. SLLN 2.
- Characteristic functions. CLT.
- Gaussian vectors.

#### 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]