One Thousand Exercises in Probability


This book contains more than 1000 exercises in probability and random processes, together with their solutions. Apart from being a volume of worked exercises in its own right, it is also a solutions manual for exercises and problems appearing in our textbook Probability and Random Processes (3rd edn), Oxford University Press, 2001, henceforth referred to as PRP. These exercises are not merely for drill, but complement and illustrate the text of PRP, or are entertaining, or both. The current volume extends our earlier book Probability and Random Processes: Problems and Solutions, and includes in addition around 400 new problems. Since many exercises have multiple parts, the total number of interrogatives exceeds 3000.

Despite being intended in part as a companion to PRP, the present volume is as self-contained as reasonably possible. Where knowledge of a substantial chunk of bookwork is unavoidable, the reader is provided with a reference to the relevant passage in PRP. Expressions such as `clearly' appear frequently in the solutions. Although we do not use such terms in their Laplacian sense to mean `with difficulty', to call something `clear' is not to imply that explicit verification is necessarily free of tedium.

The table of contents reproduces that of PRP; the section numbers for the exercises correspond to those of PRP, and all questions are numbered as they appear in the third edition of PRP. The covered range of topics is broad, beginning with the elementary theory of probability and random variables, and continuing, via chapters on Markov chains and convergence, to extensive sections devoted to stationarity and ergodic theory, renewals, queues, martingales, and diffusions, including an introduction to the pricing of options. Generally speaking, exercises are questions which test knowledge of particular pieces of theory, while problems are less specific in their requirements. There are questions of all standards, the great majority being elementary or of intermediate difficulty. We ourselves have found some of the later ones to be rather tricky, but have refrained from magnifying any difficulty by adding asterisks or equivalent devices. If you are using this book for self-study, our advice would be not to attempt more than a respectable fraction of these at a first read.

We pay tribute to all those anonymous pedagogues whose examination papers, work assignments, and textbooks have been so influential in the shaping of this collection. To them and to their successors we wish, in turn, much happy plundering. If you find errors, try to keep them secret, except from us. If you know a better solution to any exercise, we will be happy to substitute it in a later edition.

We acknowledge the expertise of Sarah Shea-Simonds in preparing the \TeX script of this volume, and of Andy Burbanks in designing the front cover, which depicts a favourite confluence of the authors.

March 2001