Synopsis for Probability

Overview

The first half of the course takes further the probability theory that was developed in the first year. The aim is to build up a range of techniques that will be useful in dealing with mathematical models involving uncertainty. The second half of the course is concerned with Markov chains in discrete time and Poisson processes in one dimension, both with developing the relevant theory and giving examples of applications.

Synopsis

Continuous random variables. Jointly continuous random variables, independence, conditioning, bivariate distributions, functions of one or more random variables. Moment generating functions and applications. Characteristic functions, definition only. Examples to include some of those which may have later applications in Statistics.

Basic ideas of what it means for a sequence of random variables to converge in probability, in distribution and in mean square. Chebychev and Markov inequalities. The weak law of large numbers and central limit theorem for independent identically distributed variables with a second moment. Statements of the continuity and uniqueness theorems for moment generating functions.

Discrete-time Markov chains: definition, transition matrix, n-step transition probabilities, communicating classes, absorption, irreducibility, calculation of hitting probabilities and mean hitting times, recurrence and transience. Invariant distributions, mean return time, positive recurrence, convergence to equilibrium (proof not examinable). Examples of applications in areas such as: genetics, branching processes, Markov chain Monte Carlo. Poisson processes in one dimension: exponential spacings, Poisson counts, thinning and superposition.

1. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (3rd edition, OUP, 2001). Chapters 4, 6.1-6.5, 6.8.
2. R. Grimmett and D. R. Stirzaker, One Thousand Exercises in Probability (OUP, 2001).
3. G. R. Grimmett and D J A Welsh, Probability: An Introduction (OUP, 1986). Chapters 6, 7.4, 8, 11.1-11.3.
4. J. R. Norris, Markov Chains (CUP, 1997). Chapter 1.
5. D. R. Stirzaker, Elementary Probability (Second edition, CUP, 2003). Chapters 7-9 excluding 9.9.