Synopsis for B12a: Applied Probability

[This option was formerly OBS3a and is paper BS3 in the Honour School of Mathematics & Statistics. Teaching responsibility of the Department of Statistics.]

Level: H-Level Method of Assessment: 1-hour written examination (3-hours if taken as a whole-unit with OBS3b)

Recommended Prerequisites:

Part A Probability. Weight: Half-unit (OSS paper code 2A72). Can be taken as a whole-unit with OBS3b.

The whole-unit (B12a and OBS3b) has been designed so that a student obtaining at least an upper second class mark on the whole unit can expect to gain exemption from the Institute of Actuaries' paper CT4, which is a compulsory paper in their cycle of professional actuarial examinations. The first half of the unit, clearly, and also the second half of the unit, apply much more widely than just to insurance models.

Overview

This course is intended to show the power and range of probability by considering real examples in which probabilistic modelling is inescapable and useful. Theory will be developed as required to deal with the examples.

Synopsis

Poisson processes and birth processes. Continuous-time Markov chains. Transition rates, jump chains and holding times. Forward and backward equations. Class structure, hitting times and absorption probabilities. Recurrence and transience. Invariant distributions and limiting behaviour. Time reversal. Renewal theory. Limit theorems: strong law of large numbers, strong law and central limit theorem of renewal theory, elementary renewal theorem, renewal theorem, key renewal theorem. Excess life, inspection paradox.

Applications in areas such as: queues and queueing networks - M/M/s queue, Erlang's formula, queues in tandem and networks of queues, M/G/1 and G/M/1 queues; insurance ruin models; applications in applied sciences.

1. J. R. Norris, Markov Chains (Cambridge University Press, 1997).
2. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (3rd edition, Oxford University Press, 2001).
3. G. R. Grimmett and D. R. Stirzaker, One Thousand Exercises in Probability (Oxford University Press, 2001).
4. S. M. Ross, Introduction to Probability Models (4th edition, Academic Press, 1989).
5. D. R. Stirzaker: Elementary Probability (2nd edition, Cambridge University Press, 2003).