An introduction to probability theory and its applications

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…ke you're more into probability modelling and machine learning than statistics in the traditional hypothesis testing sense. Besides ESL, a book I'd recommend is Bishop's Pattern Recognition and Machine Learning. It starts from the beginning of probability theory applied to computer science problems, and covers every modern topic. videolectures.net is filled with lectures on CS-flavored probability modelling and machine learning topics. The best bet is the multi-hour "tutorial" lecture series and minicourses; it may take a while to choose the right starting point. For serious stats and probability without the CS flavoring (not useful for the quick-road-to-hacking-power agenda): For classical deep stats theory, everyone I know begins with Cassela and Berger's Statistical Inference. Don't expect algorithms in this though. On the probability …

My recommendations: Hardy, A Course of Pure Mathematics. Knuth, Graham and Patashnik, Concrete Mathematics. Hoffman and Kunze, Linear Algebra (there are several free online books on Linear Algebra and Vector Spaces, Strang, Axler, Lang Halmos and Shilov are also recommended). Feller, An Introduction to Probability Theory and Its Applications. If you're not studying for exams and you have plenty of time on your hands, try the following algorithm: 1. Dive into some random chapter in the book and attempt the first exercise in that chapter. 2. Understand the problem by referring back and learning everything that might potentially have relevance to the exercise. 3. Attempt to solve exercise. 4. If unsuccessful, go to step 2. 5. If there are more exercises in the chapter, pick the next exercise and go to step 2. 6. If there are still unsolved e…

The all time classic at the level your looking for is William Feller's two volume set "Introduction to probability theory and its applications"- if you can find a copy. A good university library should have a copy. On wikipedia, it claims that Persi Diaconis (of the seven shuffles theorem fame) was motivated to finish high school after dropping out so that he could understand the contents of this book. 'nuff said. Karatzas and Shreve "Brownian Motion and Stochastic Calculus" is also a popular book among quants.