Probability Theory: A Concise Course (Dover Books on Mathematics)

"Concise" is indeed the operative word here. This book is probably not suitable as a first text on the subject, but makes an excellent review or quick reference for the topics it covers. Essentially, this text is geared toward taking someone who has - in principle - no knowledge of probability and introducing them specifically to Markov processes. There is very little attention paid to conditional probabilities, and Bayes' rule is never even mentioned. Also, this book requires no measure theory. Chapter 1 covers basic concepts: probability as relative frequency, sampling with and without replacement, binomial and multinomial coefficients. Chapter 2 is titled "Combination of Events". It introduces the idea of the sample space, and focuses on how probability interacts with set theoretic operations such as intersection and union. It ends with a proof of the First Borel-Cantelli Lemma. The third chapter introduces independence and ends with a proof of the Second Borel-Cantelli Lemma. The Borel-Cantelli Lemmas are somewhat technical results that are needed to the get the theory of Markov processes off the ground, so it's pretty clear where this book is headed early on. The proofs of both of the lemmas are very tidy. Chapter 4 is devoted to random variables. Here we find the definitions of expectation, variance, and the correlation coefficient along with Chebyshev's Inequality. Chapter 5 covers the Bernoulli distribution, the Poisson distribution, and the Normal distribution. We are also treated to the De Moivre-Laplace theorem as a stepping stone toward the Central Limit Theorem. Chapter 6 is titled "Some Limit Theorems". We are immediately provided with the proof and then statement - in that order - of the Weak Law of Large Numbers. We are then provided merely with the statement of the Strong Law of Large Numbers. This chapter then introduces Generating Functions which are used quite heavily in the remainder of the work. This chapter also introduces Characteristic Functions, which don't get much attention and concludes with the Central Limit Theorem. Chapter 7 introduces Markov Chains while chapter 8 covers Continuous Markov Processes and naturally covers the Chapman-Kolmogorov equations. Here simply called the Kolmogorov equations for the fairly obvious reason that the author is Russian. The book ends with four short appendices which introduce the reader in turn to the following topics: Information Theory, Game Theory, Branching Processes, and Optimal Control. I thought these were wonderful although obviously none of them covers very much ground. This book is actually quite delightful especially for someone who already has some background in basic probability. It does provide and good and very quick introduction to Markov processes, but it's scope of coverage of any topic is necessarily quite limited.

I love this book but can see why some people don't. It doesn't hold your hand, and it doesn't spend much time on intuition. But if you have a sense of what "mathematical elegance" is, then you will probably love this book. It gets right to the point, unlike lots of probability theory books, and is based on standard modern probability theory. Lot of worked and unworked exercises help with intuition. I did find a few typos but for the price it is excellent. Another minor flaw is that definitions and theorems are not marked as such, so occasionally you have to read carefully to see which it is. Not recommended for those with little math background, or who want a book that will "explain probability theory".

This is an excellent little book, the proverbial "slim volume." But I think you really need to have seen some probability theory already in college in say, statistical mechanics class, or a general statistics class, etc. You also need to be comfortable with integrals and power series at the college level. If you don't have this background, this is not going to work for you. However, if you do have this background, this book will really clarify and pull together a lot for you, as well as introducing you to some more advanced topics, such as Markov series. The only reason that I gave it four stars instead of five is that sometimes it is overly terse. There were times when it took me a LONG time to figure out how he got from equation A to equation B. If this were a long book, that would be intolerable. In a short book, it's OK, and even rewarding, as long as it does not get out of hand - which it sometimes does.

Though its little size, the boook has a perfect cover opf the basics of statistics; from axiomatic and intuitive definitions of statistic concepts to more advanced issues as stochastic processes, the book is a very clear and deep textbook on the matter. For me, maybe the most practical book due to its size, that permits carrying it as a travel book. Perfect demonstartions of main theorems and vey clear exposition of concepts. It should be desirable to have more exercises, because exercises are perfect to complete the theoretical parts, but it should imply a thicker book.... So, a perfect trade-off between both aspects.

At the end of each chapter we find some exercises to be solved; after each theorem the author gives its proof; there are 8 chapters(basic concepts; combination of events; dependent events;random variables;three important probability distributions; some limit theorems;Markov Chains; Continuous Markov Processes;Appendix 1-Information Theory; 2-game theory;3-branching processes; 4-Problems of optimal control;).The author offers several examples such as: sampling with (without) replacement; the optimal choice problem; Buffon's needle problem; the lottery ticket problem;radioactive decay;Brownian motion; the Poisson and Binomial Distributions; randon flow of events; one-dimensional random walk); Kolmogorov Equations, etc.. I think this book will be helpful to the students learning statistical physics and turbulence in fluids;

The presentation of the fundamental concepts is clear and is generally followed by good examples. Many of the exercises at the end of the chapter are provided with answers for self-learners that would like feedback. All-in-all this is a very good introduction to probability theory.

Arrived in 4 days. Book has no damage whatsoever. In excellent condition.

Each section begins with a helpful introduction to some concept, and then moves at a surprisingly quick pace. Many steps are explained only briefly, so reader should be familiar to concepts before-hand. Would not recommend to a beginner, particularly due to notation. Some of the more interesting topics/proofs include: Chebyshev's inequality, De Moivre-Laplace theorem, WLLN & CLT, generating gunctions (weak convergence), characteristic functions, Markov chains, information theory (w/r/t reducing uncertainty), optimal payoffs (game theory), extinction/explosion probability (branching processes), and optimal decision rules via successive corrections (optimal control).

This is a very handy book on probability theory. All the commonly used concepts and formulae are summarized in a very compact way.

Me gustó este libro, llegó bien.

I really like pretty much all Dover's serie on Mathematics and this is one of the best books in the collection. It is concise and it might require a knowledge of basic Mathematics but it covers really important topics such as random variables, limit theorems and MCMC with enough details. The book is a revised translation of Y,A Rozanov's original book. I deeply suggest to anyone who want a good introduction on the topic of modern probability theory.

Covers a lot, from basics to advanced topics. I mainly grabbed it for Markov chains, and optimal control (at appendix) which is laid out nicely

Excellent livre, ce fût un plaisir de le lire, je le recommande grandement. Dommage qu'il n'est pas disponible en français pour les plus jeunes qui souhaiteraient le lire.