Probability and random variables: A beginner’s guide.

*(English)*Zbl 0940.60003
Cambridge: Cambridge University Press. xii, 368 p. (1999).

This book is an ideal text for an introductory university course in probability. There are no special technical requirements but it is assumed that the students have knowledge in basic mathematics. The topics are carefully and well selected. All notions are well explained. The basic statements and rules are not only formulated but some of them are given with their proofs. Instructive examples illustrate both the ideas and the facts.

In Chapters 1-3 the author introduces in an attractive way notions such as random events, operations on them and rules how to calculate probabilities. This part, combinatorial in character, is a good base to go to popular discrete distributions in Chapter 4. Among them are the binomial and the Poisson distribution. The Poisson approximation and the normal approximation (Moivre-Laplace) to the binomial distribution are also considered. Then, in Chapter 5, the reader becomes familiar with discrete and continuous distributions and some of their characteristics such as expectation, variance, moments. The discussion continues in Chapter 6 on jointly distributed random variables, their conditioning, dependence. The law of large numbers for random sequences is also treated. Finally, Chapter 7 is devoted to generating functions with interesting applications to random walk and simple branching processes.

The students will find very useful the Appendices to each of the chapters giving in a summary all facts from calculus and algebra needed to understand the material. Also at the end of each chapter there is a section of exercises. Solutions and/or hints to some of them are given at the end of the book. However it is remarkable to see a large number of worked examples included in the text with all details just after introducing a new notion or fact. These examples and exercises will be of a great importance not only to students attending the lectures, but also to readers deciding to use this book for self-education. In addition to the above, the reader has the rare chance to read a large number of impressive quotations of famous scientists, writers, philosophers, thinkers.

Obviously this is a very-well written book. The content and the informal tutorial style are such that the book can be strongly recommended as an excellent text for a first course in probability. Later, based on this, other more advanced courses in probability and statistics can be offered.

In Chapters 1-3 the author introduces in an attractive way notions such as random events, operations on them and rules how to calculate probabilities. This part, combinatorial in character, is a good base to go to popular discrete distributions in Chapter 4. Among them are the binomial and the Poisson distribution. The Poisson approximation and the normal approximation (Moivre-Laplace) to the binomial distribution are also considered. Then, in Chapter 5, the reader becomes familiar with discrete and continuous distributions and some of their characteristics such as expectation, variance, moments. The discussion continues in Chapter 6 on jointly distributed random variables, their conditioning, dependence. The law of large numbers for random sequences is also treated. Finally, Chapter 7 is devoted to generating functions with interesting applications to random walk and simple branching processes.

The students will find very useful the Appendices to each of the chapters giving in a summary all facts from calculus and algebra needed to understand the material. Also at the end of each chapter there is a section of exercises. Solutions and/or hints to some of them are given at the end of the book. However it is remarkable to see a large number of worked examples included in the text with all details just after introducing a new notion or fact. These examples and exercises will be of a great importance not only to students attending the lectures, but also to readers deciding to use this book for self-education. In addition to the above, the reader has the rare chance to read a large number of impressive quotations of famous scientists, writers, philosophers, thinkers.

Obviously this is a very-well written book. The content and the informal tutorial style are such that the book can be strongly recommended as an excellent text for a first course in probability. Later, based on this, other more advanced courses in probability and statistics can be offered.

Reviewer: Jordan M.Stoyanov (Newcastle upon Tyne)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |