Probability through problems (Q5925780)

This book is a collection of exercises covering basic notions and facts from probability theory. The choice of the material and the style of presentation make the book appropriate for undergraduate university courses. It is assumed that the reader has some knowledge in algebra and calculus. The book starts with a brief survey of terminology and notations about sets, functions, derivatives and integrals. The material is divided into the following 12 chapters: 1. Modeling random experiments, 2. Classical probability spaces, 3. Fields, 4. Finitely additive probability, 5. Sigma fields, 6. Countably additive probability, 7. Conditional probability and independence, 8. Random variables and their distributions, 9. Expectation and invariance, 10. Conditional expectation, 11. Characteristic functions, 12. Limit theorems. Each chapter starts with short theoretical notes (basic definitions and statements) followed by three parts: Problems, Hints and Solutions. This structure corresponds well to established good traditions in the area. The reader is advised to try to solve independently any of the problems, and if not successful, then to look at the ``Hints'' before going to the subsection ``Solutions''. Hints are given for all problems and complete solutions to most of them. Some more difficult problems are asterisked. In general the material is well balanced and presented smoothly. This makes the book also appropriate for self-study. A short bibliography is given to indicate some further sources for those who wish to do something more in the area. Thie reviewer can make a few minor comments about the suitability of some notations and the list of references. However one thing has to be mentioned, this is the unusual way of defining the independence of \(n\) random events (Definition 7.5, page 94). Among the available books with a similar goal are e.g. \textit{T. Cacoullos}, ``Exercises in probability'' (1989; Zbl 0677.60001) containing about 500 exercises; \textit{J. Stoyanov}, \textit{I. Mirazchiiski}, \textit{Z. Ignatov} and \textit{M. Tanushev}, ``Exercise manual in probability theory'' (1989; Zbl 0661.60002) containing 777 exercises. Curiously, when preparing the present review, the reviewer has received a copy of the new book by \textit{G. Grimmett} and \textit{D. Stirzaker}, ``One thousand exercises in probability'' (Oxford, 2001). There is no doubt that any such a book including the book under review with its 435 exercises (problems) will be useful for both students and their teachers.