This reference and text offers a thorough analysis of conditional expectations and probability measures, and demonstrates their important uses in real situations. Coverage includes applications to sufficiency, Markov processes, Martingales and analysis problems.;Presenting different characterizations of conditional operators and meausures to illuminate the abstract and functional analytic nature of conditioning, this volume: reviews the general concept of conditioning; supplies Kolmogorov's formulation and its properties; provides an axiomatic method for simplifying problems as an abstraction of Kolmogorov's general formulation; explicates sufficient statistics; and illustrates difficulties in the calculation of conditional expectations for continuous multivariate distributions and provides methods for correct solutions in many cases.;This work is designed as a reference for pure and applied mathematicians, probabilists, statisticians, information theorists, physicists and communication and industrial engineers; as well as a text for graduate level students in these discipines.
Science-Math, Mathematics, Applied, Probability-Statistics,