Книга о вероятности

Несмотря на то, что я прошел несколько курсов по теории вероятностей, как в старшей школе, так и в университете, мне трудно читать документы TCS, когда речь идет о вероятности.

Кажется, что авторы работ TCS очень знакомы с вероятностью. Они волшебным образом работают с формулами вероятности и очень легко доказывают теоремы; в то время как я должен потратить несколько часов, чтобы понять, как получена одна формула и как доказываются тождества (или неравенства).

Я решил решить свою проблему раз и навсегда: хочу прочитать книгу от корки до корки.

Итак, если вас попросят предложить одну и только одну книгу по вероятности, какую книгу вы порекомендуете?

Вы пробовали эти две книги?

1. Вероятность и вычисления: рандомизированные алгоритмы и вероятностный анализ по Митценмахеру и Апфалу.
2. Рандомизированные алгоритмы Мотвани и Рагхавана

Обратите внимание, что эти две книги охватывают гораздо больше, чем просто рандомизированные алгоритмы, например, они охватывают вероятностный метод, теорию цепей Маркова, мартингалы и т. Д., Конечно, со многими приложениями в TCS. Первая книга легче читается со многими примерами, доказательства которых были детально проработаны. Вторая книга действительно классическая, не очень обновленная, но все же очень полезная. У них обоих есть много упражнений, поэтому у вас будет много материала, чтобы практиковать то, что вы изучили.

Канонический учебник для студентов по теории вероятностей остается первым курсом по вероятности Шелдона Росса. Книга является отличным справочником / освежителем для всех остальных. Независимо от того, что утверждают несколько ворчливых интернет-обозревателей, в книге четко и с сильными мотивирующими примерами освещаются все наиболее важные темы с элементарной вероятностью.

Я думаю, что решением вашей проблемы является не чтение книги вероятностей, а чтение большего количества статей в TCS.

Most papers in TCS don't actually use very advanced probability tools. Most of them use a small collection of basic and well known probability tricks. The reason you have a hard time following them is that you are not yet familiar with this bag of tricks, and many of those papers don't bother to explain these tricks because they assume the reader knows them. Some of those tricks are not taught in most probability books, at least not in the specific form they are used in TCS papers.

Another reason is that TCS papers use a slightly different terminology than the one taught in basic probability courses - e.g., in TCS papers a random variable can usually take values in $\{0,1\}^n$, while usually in probability courses random variables are defined as taking real values.

So, by reading more TCS papers, you will get more familiar with the bag of common tricks and with the terminology, and and with time they will be easier to understand.

That said, reading a book on probability is always a good idea. Among the books suggested above, I am only familiar with "Probability and Computing: Randomized Algorithms and Probabilistic Analysis" of Mitzenmacher and Upfal, and it is a very good read - in particular, it will help you get familiar with the some of the terminology and tricks used in TCS.

To add to Dai Le's answer, a more recent book by Dubhashi and Panconesi provides many examples of the use of probability in the analysis of algorithms.

Another classic of TCS/Combinatorics oriented probability is Alon and Spencer's The Probabilistic Method.

Several related topics on different SE websites:

1. Book for probability
2. Prerequisites on probability theory
3. Supplementary reading for probability theory studies
4. What book would you recommend for non-statisticians (mostly statistical books)

While I have not read any of these books, I had the luxury of taking a look at some of them. I liked the three-volume series by HPS (Hoel, Port, and Stone). It did not expect much background, and there was a clear distinction between the topics probability, statistics, and stochastic processes ( a separate volume is devoted to each topic). Moreover, each volume is rather short.

I must reemphasize that I'm not aware of the contents any of the listed books. I invite other members to comment on this post.

Several posters in this discussion recommended Feller's two volume set. A more recent and also reportedly very good textbook is Grimmett and Stirzaker. Also, here's an interesting bibliography by a professional statistician.

A very good book :

Probability by Leo Breiman

Concrete Mathematics by Knuth et al. Much of probability is figuring out the size of your universe, and from there figuring out which fraction of your universe you are interested in.

An excellent introductory probability book for computer science people is Henk Tijms, Understanding Probability, Cambridge University Press, 2nd ed., 2007. This book distinguishes itself from other introductory probability texts by its emphasis on why probability works and how to apply it.

Of the books mentioned I agree on Brieman's "Probability", Sheldon Ross' book "A First Course in Probability" The book "Probability" by Hoel, Port and Stone from their three Volume series. Most of the other books I either don't know or don't think they are appropriate. Bayesian statistics is not part of probability theory. Kai Li Chung's "A Course in Probability Theory" is the one i learned out of along with volume II of Feller's book "An Introduction to Probability Theory and its Applications" are good books that I learned from. Feller is good for heuristics and interesting problems. Chung is good for the formal mathematics. Feller and Chung can be difficult to read though, especially for self-study. Another great writer of probability books is Sid Resnick. His book "A Probability Path" is delightful to read. Neveu's "Calculus of Probability" was another book that we used in my graduate probability course.

A great book with EE slant: http://www.mhhe.com/engcs/electrical/papoulis/ a fantastic book with CS slant: http://www.amazon.com/dp/0471333417/.

Just to add to the suggestions others have given, this note by Oded Goldreich is one of the most useful ones I've found so far. It gives a lot of examples of how probability is used in various branches Computer Science. The references at the end of the book are also definitely worth looking at.

Randomized Methods in Computation: Tentative Collection of Reading Materials

This book is used for an intro probability class at MIT. http://vfu.bg/en/e-Learning/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf

Table of contents:

• Sample Space and Probability
• Discrete Random Variables
• General Random Variables
• Further Topics on Random Variables and Expectations
• The Bernoulli and Poisson Processes
• Markov Chains
• Limit Theorems