Since its first appearance, in 1982, Probability and Random Processes has been a landmark book on the subject and has become mandatory reading for any mathematician wishing to understand chance. It is aimed mainly at final-year honours students and graduate students, but it goes beyond this level, and all serious mathematicians and academic libraries should own a copy.
William Feller's introduction to the first of his classic (and revered, by some of us) books on probability theory and its applications begins with a description of probability as "a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation."
This precisely describes the approach taken by Geoffrey Grimmett and David Stirzaker. Throughout, they carefully work within a formal algebraic structure, but they do so without losing sight of the instinct we have all derived from simple gambling games, and they take every opportunity to introduce relevant, entertaining and realistic applications. It is also worth pointing out that the book is pervaded by a wry humour that makes it entertaining to read and highlights the authors' own enthusiasm for their subject.
The basic layout of the book is not very different from the first edition, but new sections have been included on simulation, Markov chain Monte Carlo methods, coupling and its applications, geometric probability, spatial Poisson processes, stochastic calculus, the Ito integral, Ito's formula and its applications, the Black-Scholes formula, networks of queues, renewal-reward theorems and their applications. While some of these inclusions may, at first sight, seem ambitious, they are beautifully and concisely presented without losing sight of their importance and applicability. In fact, they are reason enough to urge anyone who already owns a copy of an earlier edition to upgrade.
The companion book of exercises is cleverly conceived and, for the student (and, dare I say it, those of us who set examination questions), forms a perfect complement to the main text. Although it is ostensibly a self-contained book of exercises and solutions, it includes all of the exercises and end-of-chapter problems in the third edition.
Daniel Lunn is a fellow of Worcester College, Oxford.
One Thousand Exercises in Probability. Second edition
Author - Geoffrey Grimmett and David Stirzaker
ISBN - 0 19 857221 2
Publisher - Oxford University Press
Price - £.50
Pages - 438