An Introduction to Probability Theory and Its Applications, Vol. 1
– by William Feller
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Comments about this book:
Ross is elementary, and isn’t a measure-theoretic approach, and has lots of applications (e.g. to queuing theory and operations). It’s handy as a “gimme the facts” kind of book — if you want to look up common distributions and formulae you’ll find them in Ross faster than anywhere else — but it doesn’t have all the mathematical foundations you might want.
Koralov and Sinai is a measure-theory based probability course. The second half of the book has stochastic processes, martingales, etc. If you don’t know any probability at all (let’s say… haven’t seen the Bernoulli distribution derived) or if you haven’t seen measure theory, it’s probably not intuitive enough to be your first textbook. I had no complaints with the presentation; it was all straightforward enough.
Basically, I’d split the difference between elementary and advanced by using Feller; he includes EVERYTHING so you can safely skip what you know and read what you don’t.” – Posted by SarahC at http://lesswrong.com/lw/3gu/the_best_textbooks_on_every_subject/ – by Books2Learn – 2018-02-27