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An Introduction to Probability Theory and Its Applications, Volume 1
A complete guide to the theory and practical applications of probability theory An Introduction to Probability Theory and Its Applications uniquely blends a comprehensive overview of probability theory with the real-world application of that theory. Beginning with the background and very nature of probability theory, the book then proceeds through sample spaces, combinatorial analysis, fluctuations in coin tossing and random walks, the combination of events, types of distributions, Markov chains, stochastic processes, and more. The book's comprehensive approach provides a complete view of theory along with enlightening examples along the way.
509 pages, Hardcover
First published January 1, 1968
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Displaying 1 - 11 of 11 reviews
February 2, 2013
I don't think it is necessary to seek "easier" introductions to probability than this classic. The prose is quite readable, and the mathematics is accessible to a reader who has some calculus and is willing to stick with it. (I took a peek at Volume 2, and I can't claim the same for it.) The author clearly marks out what can be skipped, so you can strengthen your probabilistic intuitions without beating yourself up too much.
Amusingly, even Feller sometimes nods.
On page 10 (2nd ed, 1957), while talking about the "Distribution of r balls in n cells", he states that "the number of possible arrangements increases rapidly with r and n. For r = 3 balls in n = 4 cells, the sample space contains already 81 points". Actually it contains only n^r = 4^3 = 64 points (not 3^4 = 81).
On page 53, problem 6(b), we are to find, among others, the probability that there are 2 repetitions among 4 random digits. 2 repetitions implies 3 similar positions, which can be chosen in C(4,3) = 4 ways. The repeated digit can be chosen 10 ways, and the remaining position can be filled in 9 ways. So the number of 4-digit numbers with 2 reps is 4*10*9 = 360. The total number of 4-digit numbers is 10^4. So P(a 4-digit numbers has 2 reps) = .036, not .063 as claimed.
These are forgivable typos that don't affect the quality of the presentation.
Amusingly, even Feller sometimes nods.
On page 10 (2nd ed, 1957), while talking about the "Distribution of r balls in n cells", he states that "the number of possible arrangements increases rapidly with r and n. For r = 3 balls in n = 4 cells, the sample space contains already 81 points". Actually it contains only n^r = 4^3 = 64 points (not 3^4 = 81).
On page 53, problem 6(b), we are to find, among others, the probability that there are 2 repetitions among 4 random digits. 2 repetitions implies 3 similar positions, which can be chosen in C(4,3) = 4 ways. The repeated digit can be chosen 10 ways, and the remaining position can be filled in 9 ways. So the number of 4-digit numbers with 2 reps is 4*10*9 = 360. The total number of 4-digit numbers is 10^4. So P(a 4-digit numbers has 2 reps) = .036, not .063 as claimed.
These are forgivable typos that don't affect the quality of the presentation.
April 7, 2025
William Feller’s An Introduction to Probability and its Applications in two volumes (Wiley, Vol. I, first edition 1950 and third edition 1968; Vol. II, first edition 1966 and second edition 1971) must be accorded honorable standing as an old warhorse which anyone seriously contemplating research in probability theory ought to know. The prerequisites are minimal (for Vol. I). A knowledge of calculus at the level of Courant and John should suffice (see our review of their textbook, here). Even though nowadays professional probability theory tends to be treated as a branch of measure theory, in this first volume at least one can get away with knowing hardly any real analysis.
The philosophical level of the first introductory chapter stays minimal, almost perfunctory, just enough to frame a sensible approach to what follows for the general reader but insufficient to satisfy those who might be philosophically inclined. The following passage telegraphs nicely Feller’s own view on the matter of conceptual foundations:
We shall no more attempt to explain the ‘true meaning’ of probability than the modern physicist dwells on the ‘true meaning’ of mass and energy or the geometry discusses the nature of a point. Instead, we shall prove theorems and show how they are applied….In the opening chapters we too shall calculate a few typical probabilities, but it should be borne in mind that numerical probabilities are not the principal object of the theory. Its aim is to discover general laws and to construct satisfactory theoretical models. [p. 3]
Incidentally, we beg to differ, as far as what Feller has to say about modern physics and geometry goes. In point of fact, Bernhard Riemann’s elementary considerations in his inaugural lecture of 1854 as to what the nature of a point might be are instrumental to the modern theory of differentiable manifolds and to the field of geometry now named after him, for his reflections lead him to a recognition of the need for a metric in order to be able to measure distances in a coordinate-free manner (as captured by his slogan, Lange ohne Lage). And, for that matter, Albert Einstein’s special theory of relativity could never have seen the light of day if he had not – as Feller animadverts he ought not to have – given himself up to pondering the ‘true meaning’ of mass and energy etc.; where else could the celebrated formula E=mc² have come from? So, Feller may be right if it be a matter of puzzle solving in the mode of Kuhnian normal science but quite wrong when it comes to providing the grounds of a revolutionary discovery.
But, perhaps one should adopt a live-and-let-live attitude, for if Feller had sought to give himself over to original thought, would he then have been as equipped as he assuredly is to ‘discover general laws and to construct satisfactory theoretical models’? Thus, what to look for in this textbook is refinement of calculational technique and comprehensive coverage of the standard material. After mastering Feller one can be confident of having enough maturity to become oneself a practitioner and to initiate one’s own program of research. The pace, moreover, is gentle: for instance, the normal approximation to a binomial distribution does not appear until chap vii, the central limit theorem not until chap x, and Vol. II in full.
Subsequent chapters cover pretty much everything an advanced undergraduate should want to know about probability theory. Let us seek to draw out what is distinctive about Feller’s treatment of the subject. Chapter two concerns combinatorics. Clearly, in view of the origins of probability theory in the study of card games, an understanding of combinatorics at more than a superficial level will be most helpful to the budding probabilist. Here, Feller enters into much more than the usual permutations and combinations (as described by the binomial coefficient): partitions, subpopulations, occupancy problems, Bose-Einstein and Fermi-Dirac statistics, runs of repeated outcomes in coin-tossing experiments, the hypergeometric distribution, waiting times and (lastly) Stirling’s formula.
Chapters three and four delve into more detail on fluctuations in coin tossing and random walks. This reviewer, who is not by training an expert probabilist, found these chapters quite illuminating for he was unaware of the quality of the results attainable, for instance, by clever application of the reflection principle: impressive and enabling exact analytical statements about such things as last visits, long leads, changes of sign, maxima and first passages.
In chapter five, one returns to the exposition of standard topics: here, conditional probability and independence. The general theory is exemplified through its application to urn models, genetics and sex-linked characters – an excellent feature of the present volume, the kind of thing one would never get elsewhere. These in-depth examples illustrate how part of Feller’s objective is to teach the student to reason with more discernment about problems in probability, not just to calculate with rules.
The next four chapters (six to eight) go into great detail on an important special case, the Bernoulli trial. Feller examines the resulting binomial distribution, its central term and tails, the law of large numbers in this context, the Poisson approximation (as applied to waiting times) and the multinomial distribution. Then, in chapter seven the normal approximation to the binomial distribution, the de Moivre-Laplace limit theorem and large deviations. In chapter eight, infinite sequences of trials (as illustrated by systems of gambling). Here is where the Borel-Cantelli lemma and strong law of large numbers figure as no longer isolated statements in real analysis, but centrally in what is their proper context. The chapter concludes with a nice, rigorous proof of the law of the iterated logarithm.
After this excursion, we return to the main line of exposition in chapter nine on basic concepts associated with random variables: expectation, variance, covariance, correlation and, lastly, Chebyshev’s and Kolmogorov’s all-important inequalities. Feller goes on to prove the central limit theorem for Bernoulli trials in chapter ten but defers full proof to Vol. II (rather than proceed directly in full generality). Note: the law of large numbers reduces trivially to Chebyshev if variance is finite! When the variance is infinite, use the method of truncation to estimate lower and upper parts. Now, the next two chapters (eleven and twelve) take up special cases of random variables that permit closer analysis: integral-valued variables – for which the method of generating functions is so admirably adapted – and compound distributions, such as arise in branching processes. A few examples are computed in detail: probability of extinction and total progeny. The overall theory of branching processes offers a nice illustration of the use of generating functions.
The next two chapters (thirteen and fourteen) show well why Feller makes for such a good textbook author, for here he does not merely trick out standard formalism but provides an insightful analysis of recurrent events and renewal theory, and the problem of ruin in random walks. The theory of recurrent patterns offers illustrations of interesting types of events for which some analytical results will nevertheless be attainable (i.e., why the theory of stochastic processes, martingales and so forth isn’t almost trivial). Feller’s penetrating and dispassionate discussion of both general patterns and rare events exemplifies his pedagogical goal in the entire first volume, namely, to demonstrate by derivation of explicit formulae why our intuitive expectations about the typical behavior exhibited by random walks are often simply wrong! The last section of chapter fourteen on the generalized 1d random walk with application to sequential sampling is an example of the kind of problem that occupies probabilists but doesn’t seem important for physics (skip or skim).
The final three chapters of volume one (i.e., fifteen to seventeen) concern Markov chains, whether stationary or time-dependent. This forms a suitable conclusion to the work as a whole, as the material is at once standard and yet also nicely illustrative of technique. Feller works out the conventional results on classification, decomposition and invariant distributions. As befits his pedagogical intention, however, he will enter into some specificity on particular applications, such as to card shuffling, reflecting barriers, transient states resp. absorption, recurrence times, pure birth processes or combined birth-and-death processes, holding times and waiting line or servicing problems. Thus, what might at first appear as an abstract treatment takes on flesh as one sees how to analyze several instances of real-world applications.
To conclude, let us make the general observation that Feller’s forte is to be good on intuitive feel, certainly much better than, say, A.N. Shiryaev, whom one can come away from having plowed through all the mathematics but lacking in a handy grasp of what it all means (Probability, Springer-Verlag, second edition 1996, reviewed by us here). For example: what is a stable distribution, what does divisibility mean etc.? The reason why Feller is so superior in this respect is, namely, that he includes many sections on real-world cases. Even if one doesn’t propose to check the derivations in detail or attempt all the problems, Feller’s first volume turns out to be very much worth reading for culture and an improved feel for the real-life examples discussed throughout. Five stars.
The philosophical level of the first introductory chapter stays minimal, almost perfunctory, just enough to frame a sensible approach to what follows for the general reader but insufficient to satisfy those who might be philosophically inclined. The following passage telegraphs nicely Feller’s own view on the matter of conceptual foundations:
We shall no more attempt to explain the ‘true meaning’ of probability than the modern physicist dwells on the ‘true meaning’ of mass and energy or the geometry discusses the nature of a point. Instead, we shall prove theorems and show how they are applied….In the opening chapters we too shall calculate a few typical probabilities, but it should be borne in mind that numerical probabilities are not the principal object of the theory. Its aim is to discover general laws and to construct satisfactory theoretical models. [p. 3]
Incidentally, we beg to differ, as far as what Feller has to say about modern physics and geometry goes. In point of fact, Bernhard Riemann’s elementary considerations in his inaugural lecture of 1854 as to what the nature of a point might be are instrumental to the modern theory of differentiable manifolds and to the field of geometry now named after him, for his reflections lead him to a recognition of the need for a metric in order to be able to measure distances in a coordinate-free manner (as captured by his slogan, Lange ohne Lage). And, for that matter, Albert Einstein’s special theory of relativity could never have seen the light of day if he had not – as Feller animadverts he ought not to have – given himself up to pondering the ‘true meaning’ of mass and energy etc.; where else could the celebrated formula E=mc² have come from? So, Feller may be right if it be a matter of puzzle solving in the mode of Kuhnian normal science but quite wrong when it comes to providing the grounds of a revolutionary discovery.
But, perhaps one should adopt a live-and-let-live attitude, for if Feller had sought to give himself over to original thought, would he then have been as equipped as he assuredly is to ‘discover general laws and to construct satisfactory theoretical models’? Thus, what to look for in this textbook is refinement of calculational technique and comprehensive coverage of the standard material. After mastering Feller one can be confident of having enough maturity to become oneself a practitioner and to initiate one’s own program of research. The pace, moreover, is gentle: for instance, the normal approximation to a binomial distribution does not appear until chap vii, the central limit theorem not until chap x, and Vol. II in full.
Subsequent chapters cover pretty much everything an advanced undergraduate should want to know about probability theory. Let us seek to draw out what is distinctive about Feller’s treatment of the subject. Chapter two concerns combinatorics. Clearly, in view of the origins of probability theory in the study of card games, an understanding of combinatorics at more than a superficial level will be most helpful to the budding probabilist. Here, Feller enters into much more than the usual permutations and combinations (as described by the binomial coefficient): partitions, subpopulations, occupancy problems, Bose-Einstein and Fermi-Dirac statistics, runs of repeated outcomes in coin-tossing experiments, the hypergeometric distribution, waiting times and (lastly) Stirling’s formula.
Chapters three and four delve into more detail on fluctuations in coin tossing and random walks. This reviewer, who is not by training an expert probabilist, found these chapters quite illuminating for he was unaware of the quality of the results attainable, for instance, by clever application of the reflection principle: impressive and enabling exact analytical statements about such things as last visits, long leads, changes of sign, maxima and first passages.
In chapter five, one returns to the exposition of standard topics: here, conditional probability and independence. The general theory is exemplified through its application to urn models, genetics and sex-linked characters – an excellent feature of the present volume, the kind of thing one would never get elsewhere. These in-depth examples illustrate how part of Feller’s objective is to teach the student to reason with more discernment about problems in probability, not just to calculate with rules.
The next four chapters (six to eight) go into great detail on an important special case, the Bernoulli trial. Feller examines the resulting binomial distribution, its central term and tails, the law of large numbers in this context, the Poisson approximation (as applied to waiting times) and the multinomial distribution. Then, in chapter seven the normal approximation to the binomial distribution, the de Moivre-Laplace limit theorem and large deviations. In chapter eight, infinite sequences of trials (as illustrated by systems of gambling). Here is where the Borel-Cantelli lemma and strong law of large numbers figure as no longer isolated statements in real analysis, but centrally in what is their proper context. The chapter concludes with a nice, rigorous proof of the law of the iterated logarithm.
After this excursion, we return to the main line of exposition in chapter nine on basic concepts associated with random variables: expectation, variance, covariance, correlation and, lastly, Chebyshev’s and Kolmogorov’s all-important inequalities. Feller goes on to prove the central limit theorem for Bernoulli trials in chapter ten but defers full proof to Vol. II (rather than proceed directly in full generality). Note: the law of large numbers reduces trivially to Chebyshev if variance is finite! When the variance is infinite, use the method of truncation to estimate lower and upper parts. Now, the next two chapters (eleven and twelve) take up special cases of random variables that permit closer analysis: integral-valued variables – for which the method of generating functions is so admirably adapted – and compound distributions, such as arise in branching processes. A few examples are computed in detail: probability of extinction and total progeny. The overall theory of branching processes offers a nice illustration of the use of generating functions.
The next two chapters (thirteen and fourteen) show well why Feller makes for such a good textbook author, for here he does not merely trick out standard formalism but provides an insightful analysis of recurrent events and renewal theory, and the problem of ruin in random walks. The theory of recurrent patterns offers illustrations of interesting types of events for which some analytical results will nevertheless be attainable (i.e., why the theory of stochastic processes, martingales and so forth isn’t almost trivial). Feller’s penetrating and dispassionate discussion of both general patterns and rare events exemplifies his pedagogical goal in the entire first volume, namely, to demonstrate by derivation of explicit formulae why our intuitive expectations about the typical behavior exhibited by random walks are often simply wrong! The last section of chapter fourteen on the generalized 1d random walk with application to sequential sampling is an example of the kind of problem that occupies probabilists but doesn’t seem important for physics (skip or skim).
The final three chapters of volume one (i.e., fifteen to seventeen) concern Markov chains, whether stationary or time-dependent. This forms a suitable conclusion to the work as a whole, as the material is at once standard and yet also nicely illustrative of technique. Feller works out the conventional results on classification, decomposition and invariant distributions. As befits his pedagogical intention, however, he will enter into some specificity on particular applications, such as to card shuffling, reflecting barriers, transient states resp. absorption, recurrence times, pure birth processes or combined birth-and-death processes, holding times and waiting line or servicing problems. Thus, what might at first appear as an abstract treatment takes on flesh as one sees how to analyze several instances of real-world applications.
To conclude, let us make the general observation that Feller’s forte is to be good on intuitive feel, certainly much better than, say, A.N. Shiryaev, whom one can come away from having plowed through all the mathematics but lacking in a handy grasp of what it all means (Probability, Springer-Verlag, second edition 1996, reviewed by us here). For example: what is a stable distribution, what does divisibility mean etc.? The reason why Feller is so superior in this respect is, namely, that he includes many sections on real-world cases. Even if one doesn’t propose to check the derivations in detail or attempt all the problems, Feller’s first volume turns out to be very much worth reading for culture and an improved feel for the real-life examples discussed throughout. Five stars.
August 1, 2013
I read this classic book for my Introduction to Probability class and found it quite pleasurable. It (the first volume) gives a pretty comprehensive overview of elementary probability. The author could be verbose at times but it does not really bother me. As its name suggests, it contains multiple interesting real-world examples (albeit not today world) to illustrate how we can apply the theoretical concepts.
February 9, 2019
Introduction and purpose of Probability is explained in the simplest possible way. I liked this book for the same, However, This books slowly transits to the zone where it become too much technical/mathematical. I felt uncomfortable after sometime. It was okay as a theoretical book but I was looking for simplified practical approach.
September 18, 2017
A classic textbook of introduction to probability. It's an upper level in college or graduate school book. If you study physics or engineering, this is a book that can help to build a solid background in probability.
December 11, 2021
Good this book as a recommendation by undergrad supervisor Oscar Sotolongo. I have come back to it in several occasions. Specially to the topic of stable distributions, the equivalent of the normal distribution in the fat tails world.
May 22, 2024
Part 1 of Feller’s masterpiece! A must for any probabilist.
Read
June 13, 2010Greatly enjoyed my intro probability class but interested in plugging holes and exploring further. Heard this was the probability monogram and have high expectations.
September 11, 2015
i admit, this is as good as everything says it is.
June 29, 2024
Still the best introduction to probability theory.
Want to read
June 2, 2010Apparently an old classic -- I'm poking through in my spare time and it seems good so far.
Displaying 1 - 11 of 11 reviews