## William Feller's *Probability Theory* - Reviews 1

Below is reviews of the First Edition of Volume I. For Reviews of the Second Edition of Volume I, see THIS LINK. For Reviews of the Third Edition of Volume I, see THIS LINK. For Reviews of the First Edition of Volume II, see THIS LINK. For Reviews of the Second Edition of Volume II, see THIS LINK. |

**1. An Introduction to Probability Theory and its Applications Volume 1 (1950), by William Feller.**

**1.1. Review by: B C Brookes.**

*The Mathematical Gazette*

**35**(313) (1951), 216-217.

Dr Feller, Professor of Mathematics at Cornell University, is already well known in this country for his contributions to the mathematical theory of probability and for his special interest in stochastic processes. In this book his explicitly stated aim is to present probability theory "as a self-contained mathematical subject rigorously, avoiding nonmathematical concepts", making a "serious attempt to unify methods". The book is not, however, intended to be purely formal. Wherever possible the author indicates the practical applications of his theoretical results to problems in many fields of science and engineering. In this volume, the first of two, the author succeeds admirably in his task. He never loses sight of his primary aim, the lucid development of the mathematical theory; and though the descriptions of applications are adequate, informative and entertaining, they are never allowed to obscure the main argument. The reader always knows precisely where he stands and whither he is being led.

**1.2. Review by: David Blackwell.**

*Journal of the American Statistical Association* **46** (255) (1951), 381-382.

It is a rare event when a mathematics book appears that is admirably suitable for a textbook in an undergraduate course and at the same time is of great interest to the specialist, but Professor Feller has succeeded in writing such a book. In keeping with the modern approach to probability theory, the sample space - whose points are the possible outcomes of an experiment - is introduced as a basic concept, and it is emphasized that events are subsets of sample space and that random variables are functions defined on a sample space. Thus the reader actually finds out what a random variable is rather than merely, as in many treatments of probability, what a distribution function is. In order to avoid measure theory difficulties, only discrete sample spaces are considered. ... The style is lively and informal, and gives the reader an intuitive feeling for the methods and applications of probability theory. ... This is a beautiful book and an important one, and Professor Feller's readers await eagerly the appearance of Volume II.

**1.3. Review by: H L S.**

*Journal of the Institute of Actuaries* (1886-1994) **77** (2) (1951), 323-326.

Although [Feller's] treatment of the fundamentals is classic, or rather neo-classic, the work is distinguished by the novel idea of restricting Vol. I to discrete random variables, by a superior treatment of the laws of large numbers, and by six engagingly written chapters on random processes. Nevertheless, there is no reason to regard the work as 'cranky' in spite of its originality and there are good grounds for believing that for many years it will be *the* probability text-book *par excellence*.

**1.4. Review by: David G Kendall.**

*Journal of the Royal Statistical Society Series A (General)* **114** (2) (1951), 249-250.

This is an extremely important book, which will exercise a profound influence on the development of the subject. The author, who is now Professor of Mathematics at Princeton (though the book was written while he was still at Cornell), enjoys an international reputation for his work on the analytical theory of probability, and he has also taken an active part in the development of many of the applications. He explains in the preface that his original intention was to write a treatise on analytical methods in probability theory in which the latter was to be treated as a topic in pure mathematics, but he was persuaded by the United States Office of Naval Research to attempt the much more difficult task of writing a book which would be of equal value to the mathematician interested in the rigorous theory, and to the biologist, engineer or physicist looking only for models and techniques of value in his special field. The result is an unqualified success

**1.5. Review by: P A P Moran.**

*Biometrika* **38** (1/2) (1951), 259-260.

This is the first part of a comprehensive treatise on probability theory. It deals with probabilities defined on discrete sample spaces only, and the more difficult theory of probability in a continuous sample space is left to a second volume. By imposing such a severe restriction on subject-matter in the first volume the author has left himself free to write a most thorough and inspiring account of this part of the theory. ... This book is a treatise on probability and not on statistics, and as such must be regarded as the most important written in modern times. However, no attempt is made to provide anything like a comprehensive bibliography. At the end of each chapter is a series of exercises, many of them very interesting in themselves. From these one can apparently deduce that professors at Cornell often park their cars where parking is forbidden, play a great deal of poker and bridge, and sometimes have difficulty in finding the right key for their front doors on arriving home. The style is clear and amusing and there seem to be remarkably few misprints.

**1.6. Review by: F A C Sevier.**

*The American Mathematical Monthly* **59** (4) (1952), 265-266.

The discussions and definitions of mathematical probablility found in college algebra books reduce the theory to a pure a priori system and thereby detract from the real purpose of probability theory. Too often the problems and examples are either trivial or so detached from application as to appear ridiculous. The history of mathematics, as historians of science have shown, displays a fascinating interplay between theory and application. Certainly it is reasonable to expect that learning will be facilitated if some similar pattern is followed in the presentation of material to the neophyte. Dr Feller in this new text had such a novel view and presentation. As it should be with all basic concepts, he introduces the concept of probability with great caution by using intuitive examples before proceeding to a more rigorous and abstract definition. The use of sample space, in this volume discrete sample space, as the basis for definition is intuitively more pleasant and to this reader more elegant. Not only does this approach avoid the falseness of the usual ratio definition but it lends itself more readily to transition to statistical applications.

**1.7. Review by: D H Potts.**

*The Mathematics Teacher* **44** (1) (1951), 60.

It occasionally happens that when a first rate mathematician "stoops" to writing a text book the result is also first-rate. Such is the case in point. The author is one of the leading mathematicians of our day; his product is a highly interesting, skilfully written book which should prove to be greatly used in years to come. His treatment of the subject of probability is entirely mathematical, the speculative philosophical aspects having been properly divorced from the mathematical theory. In this volume the author has confined himself to consideration of "discrete sample spaces," thus requiring of the reader only an elementary mathematical back ground (a first course in calculus would suffice for much of the material). Problems involving more advanced mathematical techniques have been deferred to a second volume. However, such relatively advanced topics as infinite Markov chains and random walk are included. From a pedagogical point of view the reviewer thought this book excellent

**1.8. Review by: A Chapanis.**

*The Quarterly Review of Biology* **26** (3) (1951), 328.

This is a rigorous treatment of probability theory. Starting with some rather simple problems of combinatorial analysis, the text then goes on to such advanced topics as Markov chains, recurrent events, random walks, waiting time, trunking problems, and time-dependent stochastic processes. ... This is a much to difficult textbook for most students in the biological sciences, but those who can handle this kind of material should find the book rewarding. In my opinion, this is the best and most adaptable book on probability theory which has ever reached my desk.

**1.9. Review by: J Wolfowitz.**

*Bulletin American Mathematical Society* **57** (2) (1951), 156-159.

This is the first volume of a projected two-volume work. In order to avoid questions of measurability and analytic difficulties, this volume is restricted to consideration of discrete sample spaces. This does not prevent the inclusion of an enormous amount of material, all of it interesting, much of it not available in any existing books, and some of it original. The effect is to make the book highly readable even for that part of the mathematical public which has no prior knowledge of probability. Thus the book amply justifies the first part of its title in that it takes a reader with some mathematical maturity and no prior knowledge of probability, and gives him a considerable knowledge of probability with the necessary background for going further. The proofs are in the spirit of probability theory and should help give the student a feeling for the subject. ... this is a superb book, and a delight to read. The gathering together of so much material in so brilliant a manner represents a prodigious amount of labour for which the mathematical public is greatly indebted. The reviewer congratulates the author; he has set a lofty standard for would-be writers of similar books to attain.

**1.10. Review by: R Fortet.**

*Mathematical Reviews* MR0038583 **(12,424a)**

To avoid advanced mathematical concepts (measure theory, etc.) and to make the work useful to beginners, the author limits it to questions which involve only a countable sample space; but about these simple questions it addresses the most advanced problems of probability theory, many of which have not until now been exposed in a book, so that the work is of the highest interest for specialists.

JOC/EFR August 2016

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