Unit information: Applied Probability 2 in 2011/12

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Unit name Applied Probability 2
Unit code MATH21400
Credit points 20
Level of study I/5
Teaching block(s) Teaching Block 2 (weeks 13 - 24)
Unit director Dr. Leslie
Open unit status Not open
Pre-requisites

MATH 11002, MATH 11003 and MATH 11340
Co-requisites

None
School/department School of Mathematics
Faculty Faculty of Science

Description including Unit Aims

This unit will concentrate on the study of random processes, models of systems in which a random quantity varies through time. Renewal processes, Markov processes in discrete and continuous time and branching processes will be covered in detail. These provide models of, amongst other things, industrial processes, queuing systems, population growth and many other fundamental systems in the physical, biological and social sciences.

Aims

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

Syllabus

Probability spaces, continuity of probability, introduction to stochastic processes.

Probability generating function. Galton-Watson branching process with an analysis of population growth and extinction probabilities.

Poisson process. Birth and death and other continuous time stochastic processes.

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.

Introduction to martingales. Statement of the Optional Stopping Theorem and Martingale Convergence Theorem. Applications of these theorems.

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 Queueing Networks, Probability 3, Bayesian Modeling B, and also Financial Mathematics, and is relevant to other Level H/6 probabilistic units.

Intended Learning Outcomes

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.

Teaching Information

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

Assessment Information

The unit mark for Applied Probability 2 is calculated from one 2 ½ -hour examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted.

Reading and References

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

1. Grimmett, G.R. & Stirzaker, D.R. Probability and Random Processes. (OUP).

2. Taylor, H.M. & Karlin, S. An Introduction to Stochastic Modelling (3rd Ed.) (Academic Press).

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