Probability Theory

doi:10.1117/3.268105.ch1

Probability Theory

Book: Random Processes for Image and Signal Processing

Author(s): Edward R. Dougherty

Published: 1998

https://doi.org/10.1117/3.268105.ch1

Author Affiliations +

Abstract

Probability theory is concerned with measurements of random phenomena and the properties of such measurements. This opening section discusses the formulation of event structures, the axioms that need to be satisfied by a measurement to be a valid probability measure, and the basic set-theoretic properties of events and probability measures. At the outset we posit a set S, called the sample space, containing the possible experimental outcomes of interest. Mathematically, we simply postulate the existence of a set S to serve as a universe of discourse. Practically, all statements concerning the experiment must be framed in terms of elements in S and therefore S must be constrained relative to the experiment. Every physical outcome of the experiment should refer to a unique element of S. In effect, this practical constraint embodies two requirements: every physical outcome of the experiment must refer to some element in S and each physical outcome must refer to only one element in S. Elements of S are called outcomes.