probability textbooks
Has anyone compiled a moderately comprehensive list on the web or elsewhere of textbooks on probability
For students who have not been introduced to the subject before
That introduce both discrete and continuous probability distributions and their cumulative distribution functions, and include things like the Poisson limit theorem, the central limit theorem (say the former with proof and the latter not necessarily), and
That perhaps cover the simplest stochastic processes like the Poisson process or infinite sequences of Bernoulli trials?
I'm just finishing and undergrad course in Probability Theory using A First Course in Probabilty by Ross (8th Ed). it covers pretty much every topic on your list, starting with basic combinatorial principles and the basic axioms of probability and building from there. It covers all the major discrete & continuous distributions, with density, cumulative density, and moment generating functions.
The material is generally well presented, rather reliant on examples though. If you're looking for a textbook to teach/learn from, it's excellent. If you're looking for reference material, I'd keep looking for something more compact.
I am nearly finished teaching a course from Jim Pitman's book Probability; it fits this description. I think it's a good book -- it's reasonably easy to read, and has a very large number of exercises. I think having many exercises is especially important for a first probability course because especially at this level, probability is more oriented towards problem-solving than theory-building, and the only way to get practice with problem solving is to solve problems.
Two disclaimers are in order: (1) it's the book I originally learned this material from, so I may have a certain irrational fondness for it; (2) I know Jim Pitman.
I first learned probability from the great 1971 classic, Introduction To Probability Theory by Hoel, Port and Stone and the very sure hand of Professor Stefan Ralescu at Queens College. Even with the ton of probability textbooks out there now, I still think there is no better textbook for the rigorous undergraduate math major with a good calculus background. I also found the book by DeGroot and Schervish's very readable and complete, particularly for mathematical statistics. Those are my recommendations-but be leery. More then half the introductory probability textbooks today aren't worth the paper they're printed on and most of them are ridiculously expensive.