目录
5 60 本教材点评
5.3 概率论 6 本
《INTRODUCTION TO PROBABILITY》
作者：David F. Anderson, Timo Seppäläinen, BenedekValkó
出版商：Cambridge University Press
出版年：2017
ISBN：9781108415859
适用范围：本科生
推荐强度：9
作者简介：David F. Anderson，威斯康星大学麦迪逊分校的数学教授。他的研究专注于概率论和随机过程，以及在生物科学中的应用。他是三十多篇研究论文的作者，也是一本有关细胞生物学中使用的随机模型的研究生教材的作者。他于2014年荣获首届数学及其应用（IMA）数学奖，并于2016年被威斯康星大学麦迪逊分校任命为Vilas助理。Timo Seppäläinen，威斯康星大学麦迪逊分校的John and Abigail Van Vleck数学系主任。他是概率论方面的七十多篇研究论文的作者，也是大偏差理论的研究生教科书的作者。他是数学统计研究所的当选院士。他是2014年IMS大奖章的讲师，2014年国际数学家大会的邀请演讲者以及2015-2016年的Simons研究员。Benedek Valkó是威斯康星大学麦迪逊分校的数学教授。他的研究专注于概率论，尤其是在随机矩阵和相互作用的随机系统的研究中。他发表了三十多篇研究论文。他获得了美国国家科学基金会（NSF）职业奖。
书评：
This book is aimed at junior/senior undergraduates with a background in calculus (single variable, infinite series, and, for parts of the book, multivariable). It contains all the topics one would want to cover in an introductory course: the axioms for probability, classical probability (i.e., a finite number of equally likely outcomes), conditional probability, random variables (continuous and discrete being treated simultaneously), various specific distributions (normal, binomial, etc.), joint distributions, limit theorems, and so on. There is more than enough material in this book for one semester, though likely not enough for two, so an instructor has some flexibility in its use. It is a nicely written book, with clear explanations and lots of examples and exercises.
In terms of topic coverage, the scope of this book is somewhat more modest than is true of Tijms’s text. Tijms covers a number of topics that are not covered here. Some are fairly non-standard topics (e.g., Kelly betting, renewal-reward processes, law of the iterated logarithm, modes of convergence) that may not likely be covered in most introductory courses. Some instructors, however, may regret the loss of an extended discussion of Markov chains. Tijms spends about a hundred pages discussing, in two chapters, both discrete and continuous Markov chains; this book does not discuss the continuous theory at all and only spends a page or so on the discrete case.
On the other hand, the book under review spends more time than does Tijms on the topic of basic counting techniques (combinations and permutations). Tijms covers this material in only four pages of text with four worked-out examples; Anderson and his co-authors spend about three times as many pages on this topic, with about three times as many examples.
To summarize and conclude: I ended my review of Tijms’s book by saying that it should be on the short list of anybody looking for a text for an undergraduate calculus-based course in mathematical probability. At the risk of repeating myself, I must say the same about this book. Either book would make a fine textbook; I think that I would lean towards this one, if only because it seems to me to be more tightly focused on the basics and doesn’t include a lot of topics that I would view as peripheral to the course, but the Tijms book certainly has some features that, based on personal preference, might make it the preferred candidate.
本书面向有微积分基础（包括一元微积分与无穷级数，部分内容也涉及多元微积分）的高年级本科生。它包含所有概率论入门课程应当涉及的内容：概率公理、古典概率模型（即有限多个等可能样本点）、条件概率、随机变量（连续和离散情形）、一些特殊的概率分布（正态分布、两点分布等）、联合分布、极限定理等等。本书内容虽然不足以支撑两学期的课程，但对一学期的课程来说绰绰有余，以方便教师们灵活地使用。清晰的解释说明和大量的例子与习题让本书脱颖而出。
从内容来看，本书所涉及的范围比起Tijms的教材要更加“平易近人”。Tijms的教材囊括了一些本书没有的内容，而其中一些内容相当冷门（如Kelly下注问题，更新报酬过程，重对数律，随机变量的不同收敛形式），以至于大多数入门课程不会涉及。一些教师可能会为本书中Markov链相关内容的缺失感到遗憾。Tijms用了两章共约一百页的篇幅来讨论连续和离散情形的Markov链；本书则完全没有讨论Markov链的连续情形，即使是离散情形也不过寥寥几笔。
另一方面，相比Tijms的书，本书用更多的篇幅来讲述基本的计数方法（排列组合）。Tijms只用了四页和四个例题来讨论这部分内容。Anderson与他的合作者们则是花了近三倍的页数在上面，同时也提供了近三倍的例题。
总而言之，我在对Tijms教材的书评最后写到，“当你在寻找面向有微积分基础本科生的概率论课程教材时，它应当在你的候选名单中”，这句话同样适用于本书。如果单单因为它更注重基础，并且没那么多在我看来十分不必要的内容，我可能会更喜欢本书。但Tijms的书也确实有一些特别之处，使其在我个人的标准下成为了概率论教材的首选。
点评人：Mark Hunacek（爱荷华州立大学数学教授）
资料整理和翻译：孙华铱 赵旭彤
《Probability: Theory and Examples, Fifth Edition》
作者：Rick Durrett
出版商：Cambridge University Press
出版年：2010
ISBN：9780521765398
适用范围：研究生
推荐强度：9
作者简介：Rick Durrett，北卡罗来纳州杜克大学数学系教授并获得了博士学位。 1976年获得斯坦福大学运筹学博士学位。在加利福尼亚大学洛杉矶分校工作了9年，在康奈尔大学工作了25年后，他于2010年移居杜克大学。他撰写了8本著作和220余篇期刊文章， 主题广泛，并指导了超过45位博士。他是美国国家科学院、美国艺术与科学学院的成员，以及数学统计研究所和美国数学学会的会员。
书评：
There are many, many excellent texts for a graduate level course on probability. Among them are books by Resnick (A Probability Path), Gut (Probability: A Graduate Course), Pollard (A User’s Guide to Measure Theoretic Probability), Williams (Probability With Martingales), Chung (A Course In Probability Theory), and enough others to fill a shelf of my bookcase.
Two of the most popular texts are Billingsley’s Probability and Measure and Durrett’s Probability: Theory and Examples, and choosing between them is (in my opinion) a matter of taste. It is perhaps a comment on the learning process that the official text when I took this course was Billingsley, but I preferred Durrett. A few years ago, while chatting at a conference with a colleague, he mentioned that the official text when he did the course was Durrett, but that he preferred Billingsley!
I used the first edition, which was quite terse, and had many typos. Some Amazon.com reviewers also complained of the quality of the index. The book is still terse, but slightly less so — just a few more words explaining tricky parts makes this a much friendlier text than before. There are still typos, but fewer than before. Most of the ones that remain are easily avoided or pointed out by an instructor. A few require more effort, for example on page 50 the symbol ˜ is used, but only defined on page 81. I checked one of the complaints about the index, concerning the definition of Poisson distribution and it is now easy to find from the index.
These changes do not dull Durret’s wit. It is hard to believe but there were several times while reviewing the book that I laughed out loud, for example when looking for the non-existent table of the normal distribution at the back of the book.
Note that the title is Probability: Theory and Examples and that is exactly what the book contains. The theory is well developed and followed by nice examples, and then very interesting (and challenging) exercises. The examples are not fully developed applications, but rather crisp examples that illustrate the preceding theory. And yes, there are times when an exercise requires more than the immediately preceding examples and theorems. You may have to think more, and the solution may involve solutions of previous exercises — but this is what real math research is like. You never know where the answer will come from.
The book has been slightly reorganized from previous editions, with part of the measure theory appendix becoming chapter 1, and the remainder still being in the appendix.
This is not a book for beginners but a rigorous text, aimed at people who want to use probability in a profound way. As such, it makes serious demands on the reader, but yields corresponding benefits to those who persevere.
And I still like it better than Billingsley’s book.
面向研究生的概率论课程有很多优秀的教材：Resnick的《A Probability Path》, Gut的《Probability: A Graduate Course》, Pollard的《A User’s Guide to Measure Theoretic Probability》, Williams的《Probability With Martingales》以及Chung的《A Course In Probability Theory》……多到足以装满我的书架。
而Billingsley的《Probability and Measure》与Durrett的《Probability: Theory and Examples》则是最受欢迎的两本，如何选择完全依照个人喜好。我上这门课时，使用的教材是Billingsley的那本，但我更喜欢Durrett的那本。几年前，当和一位同事在会议上闲聊时，他提到他上这门课时的教材是Durrett的书，但他却更喜欢Billingsley的那本。
我曾用过本书的第一版，简洁但有很多错别字和排版错误。一些亚马逊上的读者也在抱怨索引常常出错。现在的版本依旧很简洁，虽然没有第一版那么简洁，但对那些晦涩内容的少许解释说明，让这本书读起来更轻松了。书中仍有些错别字与排版错误，但比第一版少了很多。大部分未修正的错误可以被轻易地发现或是被授课教师指出。虽然有些仍要花些功夫，比如第50页的例子用到的符号‘~’，直到第81页才被具体地定义。本书第一版的评论中有一条关于找不到泊松分布索引的投诉，而从这版书的索引中找到它已经不是难事了。
书的内容随着版次更迭不断变化，但字里行间表露出的Durrett的智慧则始终如一。不过也许你很难相信，但有好几次我看这本书的时候突然大笑起来，比如在书后找到实际上并不存在的正态分布表。
注意到本书的书名是《概率论：理论与实例》，这也正是这本书所包含的内容。优秀的例子与有趣（又富有挑战性）的习题很好地协助了理论的延伸。这些例子也许并没有被完全应用于实际，但却清楚地解释了前面的理论。而且，是的，仅靠前面的理论与例题不足以解决所有习题。你需要想更多，解决问题也许需要用到之前的习题——但这不正是真正的数学研究的样子吗？你永远不会知道从哪能找到答案。
相比前几版，本版重新编排了少许内容，比如将测度论从附录移到了第一章，而其余的仍在附录中。
这是一本严苛的教材，并不面向初学者，而是面向那些想深入学习概率论的人，因此本书对读者有着很高的要求，而那些坚持下来的人也都受益良多。
并且我仍认为本书比起Billingsley的书要更胜一筹。
点评人：Peter Rabinovitch（Research in Motion的系统架构师，概率论博士学位）
资料整理和翻译：孙华铱 赵旭彤
《Probability and Random Processes》
作者：G. R. Grimmett and D. R. Stirzaker
出版商：Oxford University Press
出版年：2001
ISBN：9780198572220
适用范围：研究生
推荐强度：9
作者简介：Geoffrey Grimmett，1976年移居布里斯托尔大学（Bristol University）担任第一任终身职位，1992年移居剑桥大学统计实验室，担任数学统计学教授，2013年10月1日当选唐宁学院硕士，2020年9月起，他被任命为海尔布隆数学研究所所长。他撰写了许多有关概率论和统计力学的研究文章，以及三本名为《渗滤》（1999年），《随机聚类模型》（2006年）和《概率论》（2010年）的研究书籍。David Stirzaker，就读于牛津大学数学研究所，曾在纯数学和应用数学的各个分支中提供过教程，例如：概率，随机过程，统计，应用分析和复杂变量的函数，参与编写了五本关于概率和随机过程的教科书，涉及从基础，入门到研究生的各个层次。
书评：
This book gives an introduction to probability and its many practical application by providing a thorough, entertaining account of basic probability and important random processes, covering a range of important topics. Emphasis is on modelling rather than abstraction and there are new sections on sampling and Markov chain Monte Carlo, renewal-reward, queueing networks, stochastic calculus, and option pricing in the Black-Scholes model for financial markets. In addition, there are almost 400 exercises and problems relevant to the material. Solutions can be found in One Thousand Exercises in Probability.
Since its first appearance in 1982, Probability and Random Processes has been a landmark book on the subject and has become mandatory reading for any mathematician wishing to understand chance.It is aimed mainly at final-year honours students and graduate students, but it goes beyond thislevel, and all serious mathematicians and academic libraries should own a copy ... the companion book of exercises is cleverly conceived and ... form(s) a perfect complement to the main text. Times Higher Education Supplement
As many reviewers have pointed out, this book may be too advanced for a beginner student, but it is truly the best for a graduate student or anyone else with solid background in Mathematics and some knowledge of Probability. When I took the course based on this book I already covered Advanced Calculus (Rudin) and college-level Probability and Statistics (DeGroot), and I just loved this book. Ten years later I still use it; just a few weeks ago I needed to familiarize myself with Brownian bridges and it took me only 10 minutes with the book to get all I need to know - the concepts, the formulas, the applications - to the extent I could explain it to my teammates.
I wouldn't recommend this book as the introduction to probability, but for a serious student of mathematics this is a great book to further the understanding of probability and prepare for advanced studies in Stochastic Calculus.
资料整理：孙华铱
《Probability: An Introduction》
作者：G. Grimmett and D. Welsh
出版商：Oxford University Press
出版年：2014
ISBN：9780198709961
适用范围：本科生
推荐强度：9
作者简介：Geoffrey Grimmett，剑桥大学数学统计教授和剑桥唐宁学院硕士。他写了许多关于概率论和统计力学的研究文章，以及关于本科生和研究生水平的概率和随机过程的三本研究书和两本教科书。Dominic Welsh，牛津大学数学系名誉教授，牛津大学默顿学院名誉研究员。他于1993年被任命为牛津大学的数学教授。他的研究领域包括组合论和复杂性理论，在这些领域他撰写了一百多篇论文。他于2005年从牛津大学退休，此后一直在新西兰和巴塞罗那担任访问职务。
书评：
This book is an updated version of the 1986 classic that I learned from. There is a new chapter on Markov chains, and a few new sections and problems, but the book still retains its concise, direct style.
A few months ago I reviewed Blitzstein and Hwang’s excellent modern Introduction to Probability, which is chock full of features to ease the student’s path. How do they compare? The targets are the same — a first course in probability for students with calculus, but not measure theory. Both cover the basics: probabilities, random variables, discrete and continuous distributions, joint distributions, transformations, moments, conditioning, basic limit theorems, inequalities, and a chapter on Markov chains. Grimmett and Welsh add a chapter on branching processes, Blitzstein and Hwang add one on Markov Chain Monte Carlo.
The main difference is in how much they hold the students’ hand. Blitzstein and Hwang try everything possible to help the student understand the material. Grimmett and Welsh present the material unaided. Blitzstein and Hwang have problems with applications to just about anything you can think of (Google’s PageRank algorithm, legal, medical, ecology cryptography, genetics, computer science, etc.), Grimmett and Welsh have only the typical probability problems (dice, cards, weather, etc.). Blitzstein and Hwang have R code and an online companion website, Grimmett and Welsh do not. Blitzstein and Hwang have about 600 exercises, Grimmett and Welsh have about 400. Blitzstein and Hwang are close to 600 pages, Grimmett and Welsh is 270.
What it comes down to, in my opinion, is that Blitzstein and Hwang is an excellent book for a wide variety of audiences trying to learn probability. Grimmett and Welsh are clearly focusing on math students — it is narrower, has fewer excursions, and is probably more difficult as a text. The material appears simple until you try to do the exercises, at which point you realize that there were many ideas contained in a few words. When you complete the exercises, you feel that you have learned something, and it stays with you. At least, those are my fond memories from twenty-five years or so ago. I have no reason to doubt that the results of working through the current edition will last a similarly long time.
本书是我在1986年读过那本的升级版，增加了关于Markov链的一章，一些新的小节和问题，但仍保持着那种简明直白的风格。
几个月前，我回顾了Blitzstein和Hwang那本很优秀的现代概率论教材《Introduction to Probability》，书的种种细节都使学生读起来更加方便。那么它和本书比起来怎么样呢？他们的定位是相同的——面向有微积分基础学生的概率论入门教材，而不需要修习过测度论。两者都涵盖了基础部分：概率，随机变量，离散和连续分布，联合分布，变换，矩，条件概率，基本的极限定理，不等式以及Markov链。Grimmett和Welsh额外添加了一章来叙述分支过程，而Blitzstein和Hwang则加入了关于Markov链Monte Carlo方法的一章。
主要的区别在于他们如何对待学生。Blitzstein和Hwang想尽一切办法帮助学生理解书中的内容，而Grimmett和Welsh则希望学生尽可能独立思考。Blitzstein和Hwang书中的问题涉及到一切你想得到的关于概率的应用（谷歌的网页排序算法，法律，医疗，密码学，生态学，遗传学，计算机科学等等），Grimmett和Welsh则只有一些经典的概率问题（骰子，纸牌，天气等等）。Blitzstein和Hwang给出了R语言代码和一个在线伴学网站，Grimmett和Welsh没有。Blitzstein和Hwang的书有大概600道练习题，而Grimmett和Welsh的书则有大概400道。Blitzstein和Hwang的书有大约600页，而Grimmett和Welsh的书差不多有270页。
总而言之，在我个人看来，对那些来自各行各业的读者，Blitzstein和Hwang的书是一本很棒的教材。Grimmett和Welsh的书则只针对数学专业的学生——它范围更小，涉猎不多，但作为教材也更难。本书看起来很简单，直到你开始尝试做习题，你会意识到只言片语中包含着无穷无尽的数学思想。当你做完习题时，你会感觉到你确实学到了一些东西，而它们也已经扎根在你思想的深处。至少，这是我25年前的美好回忆。毫无疑问，努力研习本书的此版也能为你带来差不多的收获。
点评人：Peter Rabinovitch（Akamai的高级性能工程师，在“数据科学”提出以前就从事数据科学工作。）
资料整理和翻译：孙华铱 赵旭彤
《Elementary Probability》
作者：David Stirzaker
出版商：Cambridge University Press
出版年：2003
ISBN：9780521534284
适用范围：本科生
推荐强度：9
作者简介：David Stirzaker，就读于牛津大学数学研究所，曾在纯数学和应用数学的各个分支中提供过教程，例如：概率，随机过程，统计，应用分析和复杂变量的函数，参与编写了五本关于概率和随机过程的教科书，涉及从基础，入门到研究生的各个层次。
书评：
This fully revised and updated new edition of the well-established textbook affords a clear introduction to the theory of probability. Topics covered include conditional probability, independence, discrete and continuous random variables, generating functions and limit theorems, and an introduction to Markov chains. The text is accessible to undergraduate students and provides numerous examples and exercises to help develop the important skills necessary for problem solving.
This book is perhaps the best of the “Stirzaker/Grimmett” collection of books for serious, but non measure theoretic, study of applied probability and random processes. It is the second most comprehensive of the bunch: It is somewhat gentler than their larger “Probability and Random Processes” tome, but it is also much cheaper and I think more approachable for non-mathematicians (i.e. engineers and physicists). At the same time, it is more comprehensive than all the other books of that bunch. It does require literacy and manipulation skills in the usual undergraduate calculus and linear algebra stuff, but nothing really high-powered.
You can click to see a very detailed table of contents, so I won't bother repeating too much of that here. Basically, it provides a very example-driven treatment of the material that includes all the essential stuff, from basic discrete and continuous probability and random variables, up to Markov chains and continuous parameter Markov processes, with just about all the key applications (random walks and ruin, branching, renewal, queuing, and even a bit of Weiner and Black-Scholes at the end).
The emphasis is really on problem solving, and the book is packed with worked examples and problems with solutions. The downside is that it doesn't often “come up for air” after the calculations are done, to discuss what are the important insights from all the work. But the book is already over 500 pages long, and is extremely useful for what it does contain.
It is in some ways like a Schaum's outline on steroids – and I don't mean this in a negative way at all. In fact, for its kind, I would rate this book a must-have (hence the 5 stars I gave). I would just recommend supplementing it with other material that includes more discussion of what the results mean, and what intuition can be learned from all the hard work.
这本经过充分修订与更正的教材清晰地介绍了概率论的各个方面，包括条件概率，独立性，离散和连续随机变量，生成函数，极限定理，以及Markov链的少许内容。本书即便是对本科生而言也很容易理解，同时也提供了大量的例题与习题以培养学生们解决问题的能力。
实际上，这本书可能是“Stirzaker Grimmett”系列中最好的一本，但不包含测度论，应用概率论和随机过程。它是系列中第二全面的，仅次于那本范围更广，却也更贵的《Probability and Random Processes》（编者注：这正是本篇推送所推荐的第三本书）。但我认为对非数学家（工程师和物理学家）来说，本书也更容易理解。同时，本书也远比系列中的其他教材更加全面。本书确实需要本科生级别的微积分与线性代数基础，不过这也足够了。
你可以在网上轻易地获得本书详细的目录，所以我不再在这方面重复太多。本书基本上是依据实例来介绍与讲解概率论的种种，包括从基础的离散和连续的概率与随机变量，到Markov链和含参连续Markov过程，同时也包括几乎所有重要的应用（随机漫步，赌徒破产问题，分支过程，更新报酬，排队论，甚至在最后有一些Wiener过程和Black-Scholes期权定价模型的内容）。
本书强调如何去解决问题，并附带伴有答案的例题和习题。缺点是当解决问题后，本书通常不会对从中获得的重要思想与见解作很直观的解释。但这本书已经有五百多页了，就它包含的内容来说已经很有用了。
本书通俗易懂，我给了它“必看无疑”的评价（也因此给了它5颗星）。不过我建议读者除本书外再阅读一些其他参考书，特别是那些对问题答案进行更多讨论，以及对所学内容给出更多直观解释的书籍。
——Amazon评论
资料整理和翻译：孙华铱 赵旭彤
《Probability》
作者：Jim. Pitman
出版商：Springer
出版年：1993
ISBN：9781461243748
适用范围：本科生
推荐强度：8
作者简介：Jim Pitman，加州大学伯克利分校的数学和统计学教授，自1978年以来一直在该大学工作。他曾就读于澳大利亚国立大学和谢菲尔德大学。Jim对传统的随机过程理论与数学的其他领域（尤其是组合学）之间的接口感兴趣并研究了各种随机组合对象，他目前致力于开发各种与随机分区，随机树，不可逆的合并过程以及它们的时间反转有关的思想。
书评：
This is a text for a one-quarter or one-semester course in probability, aimed at students who have done a year of calculus. The book is organized so a student can learn the fundamental ideas of probability from the first three chapters without reliance on calculus. Later chapters develop these ideas further using calculus tools. The book contains more than the usual number of examples worked out in detail.
The most valuable thing for students to learn from a course like this is how to pick up a probability problem in a new setting and relate it to the standard body of theory. The more they see this happen in class, and the more they do it themselves in exercises, the better. The style of the text is deliberately informal. My experience is that students learn more from intuitive explanations, diagrams, and examples than they do from theorems and proofs. So the emphasis is on problem solving rather than theory.
这是一本关于概率论的教材，适用于一学期的课程，针对修习过一年微积分的学生。学生可以从本书前三章中学习概率论的基本概念，而不必依赖微积分。后面的章节则使用微积分工具进一步发展概率论理论。这本书包含了比通常教材更多的详细的例子。
对于学生来说，从这样的课程中学到的最有价值的东西是如何在新的环境中发现概率论问题，并将其与标准理论体系联系起来。他们在课堂上看到的越多，在练习中自己下得功夫越多，最终的效果就越好。这本教材的风格并不是那么的正式。我的经验是，学生们从直观的解释、图标和例子中学到的东西比从定理和证明中学到的要多。因此，重点是解决问题，而不是理论。
资料整理和翻译：孙华铱 赵旭彤