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. Because of still ongoing pandemic situation, the students may choose to work on the 2020 recorded lectures and come to tutorial sessions only. Students 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 starts with two lectures on Monday 1st if November 13:15-15:00 and 15:15-17:00 in MVF-33, see full schedule in TimeEdit. The first teaching week there will only be lectures, starting from the second week, Monday 15:15-17:00 will be dedicated to tutorials. Lecture recordings from 2020 are available from this page.
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
- Chebychev 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