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.

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Lecture notes [download]
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Lecture notes with proofs [download]

Problem sets

Problem set 1. Probability distribution. CDF and PDF.

Problem set 2. Random variables and random vectors.

Problem set 3. Independence. Convolution Formula.

Problem set 4. Expected value and variance.

Attendance & Marks

Attendance list and grades.

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.

Recommended literature

  • Probability (Graduate Texts in Mathematics) 2nd Edition - Albert N. Shiryaev.
  • Introduction to probability for Data Science - Stanley H. Shan. [download]
  • Probability and Statistics for Data Science - Carlos Fernandez-Granda.
  • Introduction To Probability - Joseph K. Blitzstein, Jessica Hwang.
  • 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]