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

5.16 FUNCTIONAL VIEW OF THE VARIANCE

Based on a functional approach similar to that used to interpret the mean, we demonstrate how the variance is a measure of the width of a pdf. Since the variance is a central moment where the mean has been subtracted such that the pdf is shifted to be centered about zero, we assume the mean of X is zero and instead examine

(5.218)

Let FX(x) be standard Gaussian with parameters . Figure 5.12(a) shows a plot of the product x2fX(x) (dashed line). Since x2 and FX(x) are nonnegative, (5.218) must also be nonnegative. If the pdf is wider (larger σ), then we can expect that x2fX(x) will have greater area because x2 is a rapidly increasing nonnegative function. This is demonstrated in Figure 5.12(b) where we see that the area under the dashed line has increased for . Unlike x in the product xfX(x) for the mean, there is no zero crossing for x2: it is an even function about the origin that equally weights FX(x) on each side of x = 0 and with positive sign. We have also plotted the cumulative area

FIGURE 5.12 Functional view of the variance, showing FX(x), x2, x2fX(x), and B(x) for a Gaussian pdf. (a) and . (b) and .

(5.219)

as x is varied over the ...