# Probability 2

## Unit aims

To survey basic models of applied probability and standard methods of analysis of such models.

## Unit description

A wide range of phenomena from areas as diverse as physics, economics and biology can be described by simple probabilistic models. Often, phenomena from different areas share a common mathematical structure. In this course a variety of mathematical structures of wide applicability will be described and analysed. The emphasis will be on developing the tools which are useful to anyone modelling applications, rather than the applications themselves.

Students should have a good knowledge of first year probability and of basic material from first year analysis. As the course builds on Probability 1 it will also deepen students' understanding of the basis of probability theory.

## Relation to other units

This unit develops the probability theory encountered in the first year. It is a prerequisite for the Level H/6 units Introduction to Queuing Networks, Further Topics in Probability 3, Bayesian Modeling and Financial Mathematics, and is relevant to other Level H/6 probabilistic units.

## Learning objectives

At the end of the course the student should should:

- have gained a deeper understanding of and a more sophisticated approach to probability theory than that acquired in the first year
- have learnt standard tools for analysing the properties of a range of model structures within applied probability

## Transferable skills

- construction of probabilistic models
- the translation of practical problems into mathematics
- the ability to integrate a range of mathematical techniques in approaching a problem.

## Syllabus

- Random walks including the gambler's ruin problem and unrestricted random walks. Absorption probabilities, transience and recurrence. The Wald lemma.
- Markov chains. Examples of chains. Chapman-Kolmogorov equations. Classification of states: communicating states, period, transience and recurrence. Mean recurrence times and equilibrium distributions for irreducible aperiodic chains.
- Continuous time Markov processes on discrete state space: theory and examples (Poisson process, Birth and death process).
- Brownian motion: basic theory and properties.
- Introduction to martingales. Statement of the Optional Stopping Theorem and Martingale Convergence Theorem. Applications of these theorems.

## Reading and References

Neither of the following two books is exactly tailored to the course, but both are excellent accounts of their subject:

- Grimmett, G.R. & Stirzaker, D.R.,
*Probability and Random Processes*. (OUP). - Taylor, H.M. & Karlin, S.,
*An Introduction to Stochastic Modelling (3rd Ed.)*. (Academic Press).

**Unit code:** MATH20008

**Level of study:** I/5

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturers:** Dr Feng Yu and Dr Skevi Michael

## Pre-requisites

MATH11007 Calculus 1, MATH10003 Analysis 1A, MATH10006 Analysis 1B, MATH11005 Linear Algebra and Geometry and MATH11300 Probability 1

## Co-requisites

None

## Methods of teaching

Lectures and problems classes. Weekly exercises to be done by the student and handed in for marking.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam

**This exam will consist of five questions. All five questions will be used for assessment.*

NOTE: Calculators are NOT allowed in the examination.

Candidates may bring into the examination room one A4 double-sided sheet of notes.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.