Probability Theory: A Concise Course (Dover Books on Mathematics) by Y. A. Rozanov (PDF)

Description

This book, a concise introduction to modern probability theory and certain of its ramifications, deals with a subject indispensable to natural scientists and mathematicians alike. Here the readers, with some knowledge of mathematics, will find an excellent treatment of the elements of probability together with numerous applications. Professor Y. A. Rozanov, an internationally known mathematician whose work in probability theory and stochastic processes has received wide acclaim, combines succinctness of style with a judicious selection of topics. His book is highly readable, fast-moving, and self-contained.The author begins with basic concepts and moves on to combination of events, dependent events and random variables. He then covers Bernoulli trials and the De Moivre-Laplace theorem, which involve three important probability distributions (binomial, Poisson, and normal or Gaussian). The last three chapters are devoted to limit theorems, a detailed treatment of Markov chains, continuous Markov processes. Also included are appendixes on information theory, game theory, branching processes, and problems of optimal control. Each of the eight chapters and four appendixes has been equipped with numerous relevant problems (150 of them), many with hints and answers. This volume is another in the popular series of fine translations from the Russian by Richard A. Silverman. Dr. Silverman, a former member of the Courant Institute of Mathematical Sciences of New York University and the Lincoln Laboratory of the Massachusetts Institute of Technology, is himself the author of numerous papers on applied probability theory. He has heavily revised the English edition and added new material. The clear exposition, the ample illustrations and problems, the cross-references, index, and bibliography make this book useful for self-study or the classroom.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐”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.

⭐The quality of the content is top notch. The author approaches the subject very rigorously and the equations and proofs are great. Only problem is the book is extremely concise. You have to be either incredibly sharp to get everything in a single paragraph/example/exercise or had previous knowledge of theory of probability.In any case, as a reference book this is perfect since it’s very compact and has both the important theorems and nice proofs for each. It’s incredibly dense, covers a lot of ground, really nice complementary book for sure.

⭐Good source on probability.

⭐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;

⭐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.

⭐Very technical mathematics, here. Somewhat intelligible for intermediate math readers. Not as comprehensive as I was hoping, as it focuses on the basics.

⭐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 onintuition. But if you have a sense of what “mathematicalelegance” is, then you will probably love this book. It getsright to the point, unlike lots of probability theory books,and is based on standard modern probability theory. Lot ofworked and unworked exercises help with intuition. I didfind a few typos but for the price it is excellent. Anotherminor flaw is that definitions and theorems are not marked assuch, so occasionally you have to read carefully to see whichit is. Not recommended for those with little math background,or who want a book that will “explain probability theory”.

⭐Quick, but excellent book on probability theory.Great review or preview.

⭐Very concise, layout quite old fashioned but excellent translation, not a flashy book but full of content with a logical style. More suited to a reader with an understanding of maths than a general reader.

⭐Gostaria de destacar alguns pontos, segundo a minha opinião.Este livro do Rozanov, é de nível intermediário. Dividido em 8 capítulos e 2 apêndices, ele não é feito de forma axiomática com base em teoria da medida, mas também não apresenta um tratamento rudimentar e prolixo presente em livros básicos tais como Ross, Hogg, DeGroot e outros.Nele, as definições são apresentadas de maneira informal e motivada, porém os resultados são formalmente apresentados com indexação de Teorema, Proposição etc. Ademais, creio que o livro apresente exemplos suficientes, se considerarmos que o mesmo foi feito para ser conciso. Aliás, considerando isso, creio que haja demais: encontrei ao menos 5 exemplos por capítulo.Em cada capítulo também pude observar no mínimo 15 exercícios, todos com dicas ou respostas – mas, em nenhum deles, a resolução é apresentada.Por isso, considero que seja um excelente livro para o nível intermediário da teoria de probabilidade, ideal para quem já teve um contato com livros básicos como os citados acima por exemplo.Como leitura posterior, sugiro o livro, em nível avançado, do Achim Klenke, “Probability Theory: A Comprehensive Course”. Para graduandos ou pós-graduandos em Matemática que relegaram o estudo de Probabilidade de maneira indeterminada, sugiro fortemente esse livro para se ganhar intuição, obter uma visão geral da teoria e não perder muito tempo para tomar em seguida a leitura dos clássicos (Shryaev, Varadhan, Breiman, Chung etc) ou Klenke citado anteriormente, ou “Theory of Probability and Random Processes”, de Yakov Sinai, não tão conhecido, mas que acho muito bom.

⭐

⭐a proper technical book, well worth the money

⭐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.

⭐I am always a fan of probability maths and discrete mathematics. The intro chapter is a great foundation for someone who doesn’t understand the basics. I do recommend to go through that first chapter to get a grounding first in probability, if you skip it you can have some issues under standing the rest of the book.It’s format of some of the mathematics equations and formulae are very antiquated, as mentioned, this is why you shouldn’t skip the first chapter.

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