Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

5.1 INTRODUCTION

The expectation of a random variable is one measure of the location and center of its probability density function (pdf). It is the “expected value” in the sense that with repeated trials, the expectation would be observed on average, which is also called the mean. Other measures of the center of a distribution, such as the median and the mode, are discussed in this chapter, as well as expectations of functions of a random variable called moments. Four important moments that provide much information about the location and shape of a random variable are summarized in Figure 5.1. Basically, all expectations are integrals of a function with respect to a probability measure. Thus, it can be written as a Lebesgue integral on the abstract probability space , or as a Riemann–Stieltjes integral on for random variable X. In previous chapters, we used the Riemann integral to (i) calculate probabilities from pdfs and (ii) derive pdfs for various functions of random variables. Generally, we have assumed that the pdf is continuous or can be represented using Dirac delta functions for discrete and mixed random variables.