Expectation and Fundamental Theorems

Abstract

In the previous chapter we defined random variables and established some of their properties. Like variables in algebra, random variables have values but these values can only be determined from random experiments. To characterize all possible outcomes from such an experiment we use distribution functions, which provide a complete specification of a random variable without the necessity of defining an underlying probability space. Specifying distribution functions poses no inherent mathematical difficulties. However, in applications it is frequently not possible to determine the complete distribution function. Often a simpler representation is possible; instead of specifying the entire function we specify a set of statistical values. A statistical value is the average of a function of a random variable. This can be thought of as the value that is obtained from averaging the outcomes of a large number of random experiments as discussed in the frequency-based approach to probability of Section 2.1. In the axiomatic framework of Section 2.2 such an average corresponds to the operation of expectation applied to a function of a random variable.