Rabi Bhattacharya, Edward C. Waymire, "A Basic Course in Probability Theory"  
Springer | 2007 | ISBN: [url=]0387719385[/url] | 218 pages


IntroductoryProbability is a pleasure to read and provides a fine answer to thequestion: How do you construct Brownian motion from scratch, given thatyou are a competent analyst?

There are at least two ways todevelop probability theory. The more familiar path is to treat it asits own discipline, and work from intuitive examples such as coin flipsand conundrums such as the Monty Hall problem. An alternative is tofirst develop measure theory and analysis, and then add interpretation.Bhattacharya and Waymire take the second path. To illustrate theauthors' frame of reference, consider the two definitions they give ofconditional expectation. The first is as a projection of L2 spaces. Theauthors rely on the reader to be familiar with Hilbert space operatorsand at a glance, the connection to probability may not be not apparent.Subsequently, there is a discusssion of Bayes's rule and other relevantprobabilistic concepts that lead to a definition of conditionalexpectation as an adjustment of random outcomes from a finer to acoarser information set.