“Modeling is meant to help your decision-making process, not substitute it.” This is one of many clever pedagogical observations contained in this book, which covers the application of many probability techniques and theorems, and the concepts surrounding them, in a compelling, practical fashion. Applications range from basic probability spaces to the mathematical concept of a Wiener process, as opposed to its physical counterpart, Brownian motion. The author also works carefully through the basics of conditional probability, without submerging the reader in a morass of logical explanations. The book succinctly explains the meaning of convergence (in a probabilistic sense) and the most important limit theorems of probability, such as the strong and weak laws of large numbers.

Although some basic knowledge of linear algebra might be useful, together with some elements of partial and ordinary differential equations, the reader will find that the appendix contains some useful definitions. Overall, the book is fairly self-contained.

The exercises are not merely an application of derived formulas and techniques, but also serve to extend and clarify explanations. Some exercises are craftily used as pedagogical building blocks, to help the reader understand more complex examples or proofs. The results of many examples motivate definitions, often leading, in turn, to theorems, from which illustrative proofs are again provided.

Many proofs are used to show how certain probability techniques would work in practice, or to illustrate the meaning of some function or parameter while attempting to model a given situation.

Solutions are given for each and every one of the exercises, at least summarily, but with more than enough information to fulfill the pedagogical purpose of the book.

For clarification of some of the proofs, results, and theories, the reader is directed to other books and papers--for example, Grimmett and Stirzaker’s Probability and random processes [1], a reference in its own right--but this is only done to preserve the fluidity of the explanation, not to obscure it by omitting important facts.

This is an introductory book on probabilistic modeling that I can recommend to any student or teacher. It is not only for probability courses, but also for general mathematics, since the proofs, definitions, and examples are so beautifully intermingled and interspersed.