Short Description

The course MVE140 (GU code MSA150), Foundations of Probability Theory, gives introduction to modern probability theory with 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 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.

Learning Outcomes

On successful completion of the course the student should be able to

- identify and describe problems for which the treatment requires use of fundamental concepts and methods from Probability theory,
- apply main probabilistic theoretical or computational methods in problem solving

Reading Period

The course starts on Tuesday 25th of October 2011 (Week 43) by a lecture at 15:15 - 17:00 in MVF33 auditorium and will continue for 7 weeks. Starting from Week 44 and on there will be two lectures on Monday (in MVF31) and Wednesday (in MVF33) at 10:00 - 11:45 and a problem solving session on Monday 13:15-15:00 (in MVF33, there is no problem solving session in the starting Week 43). Follow TimeEdit link for the full timetable.

**Language**: English.

**Lecturer in charge**: Prof. Sergei Zuyev

Assessment and Examination

The grade for the course is based on the results of the written
examination which is scheduled on Friday 16th of December 2011 at 8:30am
to 1:30pm. The
result of the examination will be communicated only by email from
LADOK (and not by the student office.) This is done automatically as
soon as the results are released.

You are allowed to bring with you at the exam an English dictionary
(not Encyclopaedia type) and up to a maximum of 5 sheets of paper with your own
notes.

The following link gives more information about the examination room rules at Chalmers: Examination room instructions.

An opportunity to check the correction is required. Announcment about time and place will be posted on the course home page. The students that cannot participate at this occasion can collect and check their scripts at the student office at Mathematical scineces, Monday to Friday, at 8.30-13.00. Complaints of the marking should be written and handed in at the office. There is a form you can use (ask the personnel in the office).

Next re-examination is scheduled on Friday **January 13th 2012
at 8:30am.**. For those entitled for a second re-sit there
will be organised an oral examination in Spring 2012. Please, contact the course
organiser **prior to 15th of February 2012** if you plan to take this examination!

Course Literature

The course is built around the first half of the book

**
Geoffrey Grimmett and David Stirzaker, Probability and
Random Processes**, Oxford University Press, 3rd edition, 2001.

Syllabus

**Events and probability measure (Chapter 1):**- Probability experiment, events, sigma-albebras, probability measure
- Conditional probability, independence, product spaces

**Measurability, random variables and their distributions (Chapters 2-4):**- Random variables, distribution function
- Discrete, continuous and singular random variables, 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 (Chapter 5 and 7):**- Characteristic functions, inversion formula, continuity theorem
- Different convergence concepts for sequences of random variables
- Weak and Strong Law of Large Numbers
- Central Limit Theorem

Tutorial Problems

- For discussion on Week 44
- For discussion on Week 45
- For discussion on Week 46
- For discussion on Week 47
- For discussion on Week 48
- For discussion on Week 49
- For consultation session on Week 50

Old exams and solutions

- December 2010 examination
- January 2011 examination
- December 2011 examination
- January 2012 examination

Sergei Zuyev