MVE140 / MSA150 Foundations of probability theory Autumn 20

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 the pandemic situation, all the teaching will be carried out via zoom. You should have zoom installed on your devices and be registered as a Chalmers or a university user. 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. It also contains the link to the zoom sessions: just click the corresponding slot in the default timetable view.

Short links

Program

The schedule of the course is in TimeEdit. The first teaching week there will only be zoom lectures, starting from the second week the second Monday slot (15:15-17:00) will be dedicated to tutorials. Lecture recordings are available from this page. The tutorials will be held also on zoom in the form of a discussion, see tutorials section on how they are organised this year.

Lecture 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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Tutorials

Exercises for the following Monday tutorial will be posted each Wednesday here.

The lecturer will ask students to demonstrate their solution to selected problems, so prepare your technical setup for this kind of activity. Possibilities are:

The preferred setup: have a graphical tablet with stylus attached to your computer and write the solution in real time. A tablet with a screen, like an iPad or Microsoft Surface, can have their screen shared in zoom. Tablets without screen, like a Wacom Intuos, can be used to write on the provided Whiteboard in zoom.
Scan or photograph your solutions and download these to the device you will be using for the zoom meetings. When asked for a solution, share the screen with the corresponding scan to zoom and comment on your solution.
Have a webcam or a document camera fixed so that to show the paper with your written solution. Note that an integrated in your laptop or phone camera is usually not suitable for this because of a shaky video it would produce, especially when you want to point on something in your notes. Have your camera or the phone steadily fixed above the document instead.

Try different setups and choose the best solution that would enable you to share your solutions to zoom prior to attending the tutorial zoom session! Students who actively participate in the tutorials and demonstrate their solutions will get credits towards the final exam.

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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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