ST213 Mathematics of Random Events

Jonathan Warren

Department of Statistics

University of Warwick

The mathematical heart of probability theory is measure theory, which also lies at the heart of much of modern mathematics. This course revisits basic probability theory and develops these mathematical connections, mainly concentrating on results rather than proofs. It builds on ST111X part A and is a very useful preparation for MA359 and ST318. The ideas of this course also underlie several of the statistics courses, including ST301 Bayesian Statistics.

Aims

This course aims to provide an introduction to the mathematical ideas underlying the notion of randomness, which permeates through much of modern applied mathematics as well as statistics and probability theory. It will concentrate on the applications and examples of these ideas, rather than formal proofs (these are left to the third-year Mathematics course on Measure Theory).

Objectives

At the end of the course students will:

have been introduced to the foundations of measure theory and probability;
understand the importance of the notion of countable additivity in computing probabilities of complicated events;
understand the proper formulation of the notion of statistical independence;
understand the basic theory of integration, particularly as applied to expectation of random variables, and be able to compute expectations from first principles in simple cases;
understand the notions of convergence in probability and almost sure convergence, and the use of almost sure convergence in computation of integrals and expectations.

Syllabus

1: Events, algebras and sigma-algebras. Motivation, revision of sample space and events, algebras of sets, limit sets, sigma-algebras.
2: Probability. Measures and countable additivity, uniqueness of probability measures, Lebesgue measure.
3: Independence. Independence, Borel-Cantelli lemmas, law of large numbers for events, independence and classes of events.
4: Random variables. Simple functions and indicators. Complex random variables as limits. Distribution of random variables. Independence of random variables.
5: Integration. Integral of simple functions, andf integral of non-negative functions by monotoner limits, integrable functions, expectation of random variables, examples.
6: Convergence. Convergence of random variables, laws of large numbers for random variables, convergence of integrals and expectations, dominated convergence theorem, examples.

Lecture notes and other resources

Here is a link to summary lecture notes (in pdf format). You can make an excellent use of these as back-up to the lectures, but be aware that they were prepared several years ago: and the module this year will not exactly match them.

[Course Notes in pdf format: 424k]

(Adobe have released a free reader for pdf files which is frequently distributed on CD-ROMS of computer magazines).

In term 3 I will run two revision classes: details to be announced nearer the time.

Assessable Exercises

This course is 80% assessed by examination in Term 3 and 20% assessed by exercises which you will answer during the term in which the course is run. There are 8 worksheets that will be handed out weekly.
The following week there will be an examples class on the worksheet and a short test. Four out of the eight tests count towards the 20% assessed component. This method helps you learn during the lecture course so should:

improve your exam marks;
increase your enjoyment of the course

The worksheets, with answers included, are available as the term progresses by clicking below.

You can contact me as follows. I have office hours at regular times during weeks in term: I post these times on my office doors. Alternatively you can email me at

J.Warren@warwick.ac.uk.

Statistics' home page

Department of Statistics,

University of Warwick,
Coventry CV4 7AL, UK
Tel: +44 1203 523066
Fax: +44 1203 524532