# BSc Mathematics

Year of entry: 2022

## Course unit details:Foundations of Modern Probability

Unit code MATH20722 10 Level 2 Semester 2 Department of Mathematics No

### Overview

The law of large numbers and the central limit theorem are formulated and proved. These two results embody the most important results of classical probability theory having a large number of applications.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Probability 1 MATH10141 Pre-Requisite Compulsory
Probability 2 MATH20701 Pre-Requisite Compulsory
See above

### Aims

The course unit unit aims to

• provide the basic knowledge of facts and methods needed to state and prove the law of large numbers and the central limit theorem;
• introduce fundamental concepts and tools needed for the rigorous understanding of third and fourth level course units on probability and stochastic processes including their applications (e.g. Financial Mathematics).

### Learning outcomes

On completion of this unit successful students will be able to

• state and use fundamental inequalities (Markov, Jensen, Holder, Minkowski) and modes of convergence (almost sure, in probability, in distribution, in mean);
• state and use Fatou's lemma, monotone convergence theorem, and dominated convergence theorem;
• state and prove the law of large numbers and the central limit theorem in a variety of theoretical and applied settings;
• apply the methods of proof developed to related problems in classical/modern probability and its applications.

### Syllabus

1. Probability measures. Probability spaces. Random variables. Random vectors. Distribution functions. Density functions. Laws. The two Borel-Cantelli lemmas. The Kolmogorov 0-1 law. [4 lectures]

2. Inequalities (Markov, Jensen, Holder, Minkowski). Modes of convergence (almost sure, in probability, in distribution, in mean). Convergence relationships. Fatou's lemma. Monotone/dominated convergance theorem. [4 lectures]

3. Expectation of a random variable. Expectation and independence. The Cesaro lemma. The Kronecker lemma. The law of large numbers (weak and strong). [5 lectures]

4. Fourier transforms (characteristic functions). Laplace transforms (moment generating functions). Uniqueness theorems for Fourier and Laplace transforms. Convergence of characteristic functions: the continuity theorem. The central limit theorem. [6 lectures]

5. Brownian motion as the weak limit of a random walk. Donsker's Theorem. [3 lectures]

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• D Williams, Probability with Martingales, Cambridge University Press, 1991.
• A N Shiryaev, Probability, Springer-Verlag, 1996.
• G R Grimmett and D R Stirzaker, Probability and Random Processes, Oxford University Press, 1992.

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

### Teaching staff

Staff member Role
Goran Peskir Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.