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 |

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**

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.

**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.

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.

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

1x 2.5 hour exam

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).