MVE140 / MSA150 Foundations of probability theory Autumn 25
Short links
- Tutorial problems
- Lecture recordings (2020)
- Lecture notes (2021)
- VLE
- Exam January 2023 solutions
- Exam January 2024 solutions
- Exam January 2025 solutions
Aim and learning outcomes
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The course MVE140 (GU code MSA150), Foundations of Probability Theory, gives an introduction to modern probability theory with an emphasis on mathematical background. The measure-theoretic axiomatics proposed by Andrey Kolmogorov in 1932 has made Probability a rigorous science and enormously influenced all further developments of the subject. The course is intended for Master's students with good general mathematical knowledge; no other prerequisites are assumed, though familiarity with functional analysis, Laplace transforms and measure theory will be a great advantage. The students also have access to Year 2020 recorded lectures and may also register at the Virtual Environment platform (VLE) using their university email. The VLE contains elementary exercises on Probability to refresh your previous knowledge.
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Andrey Nikolaevich Kolmogorov (1903-1987) |
On successful completion of the course, the student should
- be able to identify and properly formulate probabilistic models for real-life phenomena;
- have a deep understanding of the grounds of probability and its relations to measure theory, set theory, and Lebesgue integration;
- know about the main probability distributions, their properties and range of applications;
- have an advanced understanding of dependence and conditioning;
- have solid competence in carrying out analytical probability calculations, including the use of transforms.
Teachers
Examiner: Sergei Zuyev
Lecturer: Sergei Zuyev
Program
The course starts with two lectures on Monday 3rd of November 13:15-15:00 and 15:15-17:00 in MVF-33, see full schedule in TimeEdit. On Wednesday 5th November, 10:00-11:45, we have our first tutorial. Starting from the second teaching week, Mondays, 15:15-17:00, will be dedicated to tutorials, and the lectures will be held on Mondays, 13:15-15:00 and on Wednesdays, 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
- Chebyshov 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.
Assessment
The grade for the course is based on the results of a written examination. To pass the course, a mark of at least 40% should be obtained (50% for PhD candidates).
In the Chalmers Student Portal you can read about when exams are given and what rules apply to exams at Chalmers.
Before the exam, you must sign up for the examination.
At the exam, you should be able to show valid identification. You are allowed to use a dictionary (to and from English), a non-programmable calculator and up to a maximum of one double-sided page of your own hand-written notes.
The main exam is scheduled for Saturday, the 17th of January 2026, in the morning.
The re-exam is scheduled for Friday, the 10th of April 2026, in the morning.
Use of AI tools
During your studies, you are free to use AI tools to support your learning. During the exam, the use of any kind of external help, including AI tools, is not allowed.
Course evaluation
At the beginning of the course, at least two student representatives should have been appointed to carry out the course evaluation together with the teachers. The evaluation takes place through conversations between teachers and student representatives during the course and at a meeting after the end of the course, when the survey result is discussed and a report is written.
Guidelines for Course evaluation in the Chalmers student portal.
Student representatives
Reference literature:
Geoffrey Grimmett and David Stirzaker. Probability and Random Processes, Oxford University Press, 3rd edition, 2001. ISBN-10: 0198572239, ISBN-13: 978-0198572220
NB. The 4th edition of this book is available from here, but the course uses the 3rd edition above.
English-Swedish mathematical dictionary.
Also recommended for the topics related to measure theory:
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