Mathematical Foundations of the Calculus of Probability

Jacques Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco. Random variables and the associated probability theory and conditional probability theory are surprisingly relevant, especially now with computing, in the real world. That the set of real valued random variables X such that E[X^2] is finite forms a Hilbert space, e.g., is complete, is astounding. So is the martingale convergence theorem. Erich Fromm, The Art of Loving. Heavily about how what people do is in response to the anxiety they feel from the realization that alone they are vulnerable to the hostile forces of nature and society.

"Can you please suggest on-line resource to learn great beginning and advanced probability concepts?" Yup! The subject at the level you ask is a major topic in applied math but is not very popular in US universities. So, you will not get many good answers. In particular, the 'computer science' community, with 'machine learning' and 'artificial' this, Bayesian that, likely won't have good answers. Your "on-line" part is asking a bit much; I can give you references to books but not all on-line. There may be some PDF files on-line, from TeX, that have such material; try some Google searching with the keywords used here. The intuitive foundations of probability went back to gambling. 'Probability' is a field of 'applied' math and as such is well defined: About 100 years ago, E. Borel student H. Lebesgue invented 'measure theory' which essentially 'rewrote' classic calculus, especially the part about integration. For the simple cases, what Lebesgue did gets the same numerical values as the classic Riemann integral. The difference is that in theoretical work Lebesgue's integral is much more general and much better 'behaved'.…