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Hi guys, I am looking for a good book to study probability. My advisor suggested the "Probability" by Leo Breiman. I am reading it now, it seems rather a dense book, so I would like to ask you guys advice on which book you guys often start with for Probability

thanks you guys for your time

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It's hard to go wrong with the two-volume set by Grimmett and Stirzaker: vol. 1 is the main text and vol. 2 are the exercise solutions. I think of this as a modern counterpart to Feller. It focuses on problems and is quite accessible, but introduces all the basic ideas that any serious student will have to master. If you are doing mathematical finance you can't go wrong with Williams' Probability with Martingales. It includes a section on a discrete version of Black-Scholes and will provide solid preparation for the real stuff. –  Steve Huntsman Apr 5 '10 at 3:08
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Steven, can you click "edit" and then check the box marked "community wiki"? This is the mode we like to use for questions that don't have a single right answer. Thanks. –  Tom Leinster Apr 5 '10 at 5:22
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11 Answers 11

up vote 6 down vote accepted

I would definitely go for "Probability" by Jim Pitman. It is a very good book for learning Probability Theory, one of the best text books I have encountered in my studies.

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Williams' book Probability with Martingales is short, thorough and fun, and will be a good introduction to the kind of probability theory used in finance.

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Can't upvote this enough. Williams teaches you how to think probabilistically without too much technical machinery, so this is a great book even if you have no interest in mathematical finance. –  Tom LaGatta May 11 '10 at 1:53
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Rick Durrett's book "Probability: Theory and Examples" is a very readable introduction to measure-theoretic probability, and has plenty of examples and exercises. This is the second text that I learned probability theory out of, and I thought it was quite good (I used Breiman first, and didn't enjoy it very much). As a bonus, there is a free .pdf version of the 4th edition (which will be published in a few months) available on his website for the time being.

A recent text covering similar material (which I admit I haven't read that fully) which looked good on a quick reading was "Probability Theory: A Comprehensive Course" by Klenke. It has a very nice selection of modern topics.

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I'm surprised to hear Durrett's book called "very readable". While it may be very readable for you, I don't doubt that at all, for me it was quite hard to get anywhere. I think if you have a STRONG background in analysis and some exposure to concrete probability (like an upperdiv stat class or something) Durrett's book might be easier. –  Michael Hoffman May 14 '10 at 21:01
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Grimmett and Stirzaker is a very good book,but it's quite difficult for the average student. The students at Cambridge who trained on these books are some of the best in the world. The best all around probability text for serious undergraduates right now is (forgive the pun) probably PROBABILITY AND STATISTICS,3ed edition,by DeGroot And Sharivich.Most undergraduate textbooks aren't comprehensive enough to cover important functions like the gamma and beta families of distributions.This one does and does it very carefully and rigorously without going too far into the theoretical side. I second the suggestion of Hoel,Port and Stone from jamie above.It's terrific and mathematically perfect.I first learned probability from it and the wonderful lectures of Stefan Ralescu. My one complaint with it is that it's ridiculously expensive.That's why I suggested DeGroot and Sharevich instead.It covers the same material and more and at a better price.If you can borrow HPS,though-by all means,please do so.

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I really enjoyed this unfinished book by Rota and Baclawski on probability and random processes. It is extremely well written (like everything else by Rota) and covers a number of interesting topics like Markov processes, entropy and information, and Brownian motion.

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There is a very good book available for free from the Scuola Normale Superiore of Pisa: Introduction to Measure Theory and Probability.

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I learned from this book by Hoel, Port and Stone. I didn't find it very dense. I thought it had a lot of good examples. If you're getting bogged down in the first 4 chapters of Breiman's book, it might be a good introduction before you learn some of the deeper stuff.

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I agree with your prof on this one, Breiman is a good book and very readable within this field, though someone above said Durrett was very readable... I suppose it is a matter of taste to some degree. If you're looking for something that's less dense than Breiman, I would highly recommend avoiding Durrett.

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HPS is much cheaper from amazon.co.uk

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I am going to shameless plug Erhan Cinlar's GTM volume Probability and Stochastics. It is maths oriented, but it doesn't assume too much (since it was based on a course taught in the Operations Research department in the engineering school), so the first thing it does is measure theory and starts from there. And it also has lots of nice exercises.

I am a bit sad that for the final published version the author went back and removed most (if not all) of the wry remarks on the history of mathematics.

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A few years ago I wrote my own `fast-moving' introduction to probability theory which may help you to see the route ahead when reading a book like Williams's Probability With Martingales.

You'll find the PDF article in my blog post here.

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