This is an excellent introductory book on random variables, with a wealth of examples and exercises. No knowledge of measure theory is assumed, since the intuitive definition of a probability space is sufficient for the purpose of the book, which is to provide the reader with a thorough understanding of the main techniques and results of probability theory. The advantage of this approach is the plain, concrete flavour of the mathematics involved; the drawback, if any, is that some more advanced developments, such as martingales and stochastic integration theory, are probably out of reach.
The material is very well organized, and covers all the classical topics: conditioning, transforms, convergence, order statistics and stochastic processes. The final chapter is devoted to a detailed analysis of the Poisson process, as an application of the notions acquired earlier in the book and at the same time as an introduction to a further study of the theory of stochastic processes. Some more advanced topics, e.g. martingales, are also touched upon, still in an elementary, down-to-earth way. Also, a few topics in statistics, e.g. regression and predictors, are described.
What are the prerequisites for this book? A course on linear algebra, on basic analysis, and (perhaps) a first course on elementary probability theory.
The text is remarkably well written, mathematically and aesthetically; layout and fonts make it a pleasant reading, and the examples are often enlightening. I think it will be a valuable support for students and instructors and it should definitely find a place in every good library.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at firstname.lastname@example.org.