This is a second edition, revised and expanded, of an introductory textbook on probability and statistics. The author uses the word stochastics to mean “the science of the rules of chance”. He views the subject as a comfortable marriage between the rigor and precision of mathematics and the world of applications with all of its inherent uncertainties.

The book is based on a course for second year mathematics students that the author taught at the University of Munich. He also recommends the book to computer science students interested in the mathematics behind applications of stochastics.

Probability and statistics are presented as two separate and quite distinct parts of the book — probability in the first six chapters, statistics in the following six. Although the topics are largely standard, the treatment here — especially with statistics — is markedly different than in a typical probability-and-statistics undergraduate course in the United States. Development of the material is very systematic but it is pitched at a fairly high level of sophistication. There is a good deal of emphasis throughout on theorems and proofs and a de-emphasis of data analysis in statistics.

Sample spaces are introduced right from the beginning via σ-algebras and probability measures are defined shortly thereafter. This happens with very little prior discussion of discrete probability. Some exposure to probability at a less formal level would be a desirable prerequisite.

The early chapter on standard stochastic models illustrates both the strengths and the shortcomings of the book. The author provides examples of standard probability distributions (uniform, Poisson, waiting time and normal) as well as descriptions of urn models with and without replacement. Although the writing is generally clear enough, the examples are often too terse, too limited in scope and stingy with details. This is problematic for an introductory textbook unless the book is really intended as a kind of advanced introduction for especially capable students. The rest of this book follows this pattern. A student could complete the statistics portion of the book and have a reasonably good grasp of the principles of introductory statistics, but with only a very limited idea how to approach data analysis.

The book has a generous number of exercises. In spirit they match the text: some require proofs, others aim toward applications, and relatively few are computational and fewer still are routine. Sketches of solutions to selected exercises are provided in an appendix

It is difficult to see how this text would fit a course in the U.S. It is too advanced for most undergraduates, but the scope is probably too narrow for a graduate class.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.