Probability: The Science of Uncertainty: With Applications to Investments, Insurance, and Engineering

This is a textbook on basic probability theory at the advanced undergraduate or beginning graduate level. It is aimed primarily at students of actuarial science. The “engineering” in the book’s subtitle, with Applications to Investments, Insurance and Engineering, is a stretch unless it means financial engineering. There are very few applications to any other kind of engineering.

The emphasis throughout the book is to develop understanding of random variables and probability distributions as they are applied in investment, insurance, and the analysis of risk. The first two chapters provide a gentle introduction and offer four extended examples to motivate key concepts. Chapter 3 presents the aspects of classical probability dealing with combinatorics, sets and counting.

The author then goes on in a lengthy Chapter 4 to give a very nice detailed treatment of random variables and probability distributions. Beyond the usual topics considered at this level, he includes survival functions, hazard functions and moment generating functions.

Chapters 5 and 6 take up special distribution functions (discrete and continuous, including binomial, Poisson, negative binomial, geometric, exponential, gamma. Pareto, DeMoivre, normal, log-normal and beta). The author also illustrates these distributions in applications that include insurance, finance, quality control, queueing theory, and product lifetime.

The following three chapters examine transformations and sums and products of random variables, as well as mixtures and compound distributions. Here too the author provides a very clear and very thorough exposition of the basic ideas and techniques. The final chapter focuses on the Markowitz portfolio selection model, a seminal piece of modern investment theory.

This book is one of several recommended by actuarial societies as a means of preparing for an actuarial exam in probability. It is clear, thorough, and pleasant to read. Even for non-actuarial students, the author’s treatment of probability distributions and random variables is definitely worthwhile. The strong emphasis on insurance and finance in examples and applications might put off some readers, but the text could be supplemented with other scientific and engineering applications. The only prerequisite is single-variable calculus.

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.