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

• Tutorial problems
• Lecture recordings (2020)
• Lecture notes (2021)
• VLE
• Exam January 2023 solutions
• Exam January 2024 solutions

Program

The course starts with two lectures on Monday 30th of October 13:15-15:00 and 15:15-17:00 in MVF-33, see full schedule in TimeEdit. On Wednesday 1st of November 10:00-11:45 we have our first tutorial. Starting from the second teaching week, Monday 15:15-17:00 will be dedicated to tutorials and the lectures will be held on Monday 13:15-15:00 and on Wednesday 10:00-11:45.

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

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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. 

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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:

Sheldon Axler. 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

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