Statistics do not lie, nor is probability paradoxical. You just have to have the right intuition. In this lively look at both subjects, David Williams convinces mathematics students of the intrinsic interest of statistics and probability, and statistics students that the language of mathematics can bring real insight and clarity to their subject. He helps students build the intuition needed, in a presentation enriched with examples drawn from all manner of applications, e.g., genetics, filtering, the Black–Scholes option-pricing formula, quantum probability and computing, and classical and modern statistical models. Statistics chapters present both the Frequentist and Bayesian approaches, emphasising Confidence Intervals rather than Hypothesis Test, and include Gibbs-sampling techniques for the practical implementation of Bayesian methods. A central chapter gives the theory of Linear Regression and ANOVA, and explains how MCMC methods allow greater flexibility in modelling. C or WinBUGS code is provided for computational examples and simulations. Many exercises are included; hints or solutions are often provided.

• Many fundamental concepts from statistics are treated; probability is treated as probabilists think of the subject. Modern computing methods are used throughout and code often provided • Difficulties in statistics are faced honestly, ample discussion of so-called paradoxes • Quantum probability and quantum computing are treated in some depth

### Contents

Preface; 1. Introduction; 2. Events and probabilities; 3. Random variables, means and variances; 4. Conditioning and independence; 5. Generating functions and the central limit theorem; 6. Confidence intervals for 1-parameter models; 7. Conditional pdfs and multi-parameter Bayesian statistics; 8. Linear models, ANOVA etc; 9. Some further probability; 10. Quantum probability and quantum computing; Appendix A. Some prerequisites and addenda; Appendix B. Discussion of some selected exercises; Appendix C. Tables; Appendix D. A small sample of the literature; Bibliography; Index.

### Reviews

'David Williams is a very distinguished mathematician with an enthusiasm for the subject which lights up the book. The book should be read, and the contents pondered on, by everyone who teaches 'second courses' on probability or statistics in a mathematics degree; any good student on such courses would surely be excited by the book.' LTSN Newsletter

'In the Preface the author warns that the book has many unusual features: this is why the book is so interesting. The book is a rich and enjoyable source of ideas, motivations and examples, which can be used by teachers of probability and statistics.' EMS

'David Williams, an author well known in the probabilists' community, has written several books which have had an important influence from the moment they were published. In summary, while progressing through the book, the reader evolves from elementary questions of heads and tails toward a global vision of probability and statistics, with the author vowing to reconcile these two domains, which have separated, then divorced. It is a goldmine of information in both domains. Bravo, David, for this new tour de force! Many readers will be looking forward to your next volume, but this should not distract you from l'art d'etre grand-père, especially in the bicentennial year of Victor Hugo.' The Mathematical Intelligencer

'I think this is a must-read book for all who will eventually work and teach in probability or statistics because it affords an enjoyable, accessible, and important perspective about their interrelationships.' Computing in Science & Engineering

'… the book?s unusual approach and entertaining style results in an introduction to statistics which is unusually accessible to an applied mathematics audience … a very readable book, from which I learned a great deal and into which I have been continuously dipping since I finished it. I would recommend it to anyone, from final year undergraduate, to an established researcher in nonlinear science. They might be surprised how interesting statistics can be, and perhaps respond more positively next time they are asked to fit data to a model.' Nonlinear Science News