ABSTRACT
The primary focus of this chapter is expectation. The chapter starts with definitions for expectation and identifies the construction of expected measures as functions of random variables that serve to characterize the distributions on those random variables. Both discrete and continuous cases are treated and most of the development addresses the mean and variance of the distributions. The key properties of expectation are enumerated and explained with emphasis on the linearity property. The discussion is then extended to random vectors with particular attention to covariance. Then the natural extension to conditional distributions is presented. The idea of functions of random variables is then generalized to describe the construction of any meaningful sort of function of a random variable or of a random vector. The use of the Jacobian to obtain multidimensional functions is explained and then the application of expectation to functions of random variables and vectors is presented. Finally, as the most commonly used function is the summation, the chapter is completed with the construction of distributions on sums of random variables.
