A Basic Course in Probability Theory
Description
Introductory Probability is a pleasure to read and provides a fine answer to the question: How do you construct Brownian motion from scratch, given that you are a competent analyst?
There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set.
Reviews / Votes
From the reviews:
"Bhattacharya (Univ. of Arizona, Tucson) and Waymire (Oregon State Univ., Corvallis) write to provide the necessary probability background for studying stochastic processes. For students exposed to analysis and measure theory, the book can be used as a graduate-level course resource on probability. . Every chapter ends with a set of exercises, including numerous solved examples. Appendixes explain measure theory and integration, function spaces and topology, and Hilbert spaces and applications to measure theory. List of symbols. Summing Up: Recommended. Graduate students; faculty and researchers." (D. V. Chopra, CHOICE, Vol. 45 (7), 2008)
"The mentioned prerequisites are exposure to measure theory and analysis. Three appendices (29 pages) provide a brief but thorough introduction to the measure theory and functional analysis that is needed. . This well-written book is full of wonderful probability theory." (Kenneth A. Ross, MathDL, February, 2008)
"The book provides the fundamentals of probability theory in a measure-theoretic framework ... . is suitable for advanced undergraduate students and graduate students. The material is presented in a very dense and concise way and each chapter includes a section with exercises at the end ... . Thus the book may be used very well as a reference text and companion literature for a lecture course ... ." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1138 (16), 2008)
"This book is a self-contained exposition of various basic elements of probability theory. It is suitable for graduate students with some background in analysis, but may also serve as a quick reference for more experienced readers. . Overall, this book is quite rich and very pleasant to read . ." (Djalil Chafaï, Mathematical Reviews, Issue 2009 e)
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