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

5.8 FUNCTIONAL VIEW OF THE MEAN

The expectation is also called the mean of random variable X: it is one measure of the center and location of the pdf. We would like to provide more insight about the mean by viewing it simply as the area of the product of two functions. Observe that the integrand in (5.53) is the product of the pdf FX(x) (which, of course, is always nonnegative) and x which is an odd function about the origin. Let FX(x) be the standard Gaussian pdf with parameters . Figure 5.4(a) shows plots of x and FX(x) superimposed, as well as their product for . Since g(x) = x is an odd function, it inverts FX(x) in the product xfX(x) for x<0. When the result is integrated, positive and negative areas of xfX(x) cancel each other; they cancel exactly when because FX(x) is an even function about the origin. We also show the cumulative area under the product:

FIGURE 5.4 Functional view of the mean, showing FX(x), x, xfX(x), and A(x) for a Gaussian pdf. (a) and . (b) and .

(5.55) ...