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. All the students are required to register at the Virtual Environment platform VLE  using the university email.  The VLE contains elementary exercises on Probability to refresh your previous knowledge and the communication means.

Program

The schedule of the course is in TimeEdit. The first teaching week there will only be lectures, starting from the second week the second Monday slot (15:15-17:00) will be dedicated to tutorials.

Content

(the references are given to the Grimmett-Stirzaker's book 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

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Recommended exercises

Exercises for the following Monday tutorial will be posted each Wednesday here and on the VLE site.

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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:

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

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