## Course syllabus

This page contains the program of the course: lectures and exercise tutorials. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM. The students have also access to Year 2020 recorded lectures and may also register at the Virtual Environment platform (VLE) using the university email. The VLE contains elementary exercises on Probability to refresh your previous knowledge.

**Short links**

### Program

The course was planned to start with two lectures on Monday 31st of October 13:15-15:00 and 15:15-17:00 in MVF-33, see full schedule in TimeEdit. But** **due to sad personal circumstances, the Monday 31/10 lectures are cancelled. The nearest lecture is planned for Wednesday 2nd of November 2022 at 10am. **Please, work out the contents of the first two lectures by watching the 2020 recorded lectures 1-7 available from this page and/or by studying Lecture Notes from 2020 (link above) and/or reading Chapter 1 in the course book by Grimmett and Stirzaker mentioned below. **Thank you for your understanding! Starting from the second teaching week, Monday 15:15-17:00 will be dedicated to tutorials.

**Course content**

*(the references are given by Grimmett-Stirzaker's book cited below)*

**Events and probability measure***(Chapter 1 without Completeness in Ch. 1.6)***:**- Probability experiment, events, sigma-fields, probability measure
- Conditional probability, independence, product spaces

**Measurability, random variables and their distributions***(Chapter 2 without 2.6, Chapter 3 without 3.9, 3.10, 4.1-4.9)***:**- Random variables, distribution function
- Discrete, continuous and singular random variables, the probability density function
- Random vectors, independence
- Expectation, variance, covariance and their properties
- Chebychov and Markov inequalities, Borel-Cantelli lemma
- Conditional distribution and conditional expectation

**Analytic methods and limit theorems***(Ch. 5.7-5.9, 5.10 up to and including Th. 4, 7.1, 7.2 without proofs):*- Characteristic functions, inversion formula, continuity theorem
- Different convergence concepts for sequences of random variables
- Weak and Strong Law of Large Numbers
- Central Limit Theorem

**Tutorials**

#### Exercises for the following Monday tutorial will be posted each Wednesday here. The students who actively participate in the tutorials and demonstrate their solutions will get credits towards the final exam. Details will be explained at the first lecture.

**Reference literature:**

**Geoffrey Grimmett and David Stirzaker. Probability and Random Processes, **Oxford University Press, 3rd edition, 2001. ISBN-10: 0198572239, ISBN-13: 978-0198572220

Also recommended for measure related topics:

* Measure, Integration & Real Analysis,* Springer, 2020 - freely available here

**Marek Capiński and Ekkehard Kopp. Measure, Integral and Probability**, Springer, 1999. ISBN-10: 1852337818, ISBN-13: 978-1852337810