MATH20712 - 2008/2009

 

Unit specification

Aims

The programme unit aims to enable students to develop some understanding of the way that stochastic processes evolve in time, to become familiar with some simple techniques which help in their study, and to experience some real life applications of stochastic processes.

Brief description

The course introduces some simple stochastic processes, that is phenomena which evolve in time in a non-deterministic way. It applies the techniques developed in Probability and Statistics 1 and 2 together with the use of generating functions (or power series) to tackle problems such as the gambler's ruin problem, or calculating the probability of the extinction of certain populations.

Intended learning outcomes

On completion of this unit successful students will be able have a good grasp of basic concepts, techniques and results associated primarily with the elementary theory of simple random walks, branching processes, and renewal processes.

Future topics requiring this course unit

This course leads naturally to third and fourth level course units on stochastic processes. The models discussed serve as nice examples for the general theory of stochastic processes.

Syllabus

1. Review of conditional probability, probability distributions, random variables, means and variances. [2 lectures]
2. Independent random variables. Sums of independent identically distributed random variables. [1]
3. Probability generating functions and their application to sums of independent random variables and random sums. [2]
4. Random walks. Recurrence and transience. Gambler's ruin problem. [7]
5. Branching processes. The size of the nth generation and its probability generating function. The probability of extinction. [6]
6. Renewal processes. The counting processes and occurrence time processes. Renewal equations and real life applications including traffic flow. [6]

Textbooks

• G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, Oxford University Press, 2000.
• S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, Academic Press, 1975.

Learning and teaching processes

Two lectures and one examples class each week. In additional students are expected to do at least fours a week of private study on this course unit.

Assessment

Coursework; Weighting within unit 20%

2 hours end of semester examination; Weighting within unit 80%

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Arrangements