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Applied Probability 2 (MATH 21400)

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Administrative Information

  1. Unit number and title: MATH 21400 Applied Probability 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1995-6
  6. Unit Organiser: David Leslie
  7. Lecturer: Dr David Leslie and Dr Feng Yu
  8. Teaching block: 2
  9. Prerequisites: MATH 11007 Calculus 1, MATH11006 Analysis 1, MATH 11005 Linear Algebra & Geometry and MATH11300 Probability 1

Unit aims

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

General Description of the Unit

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

Teaching Methods

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

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

Assessment Methods

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.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40. 

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.

Texts

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).

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.