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

5.22 FUNCTIONAL VIEW OF SKEWNESS

We can use a functional approach to show that skewness is a measure of the asymmetry of a distribution. Recall for a pdf that is symmetric about the mean, all odd central moments are zero; this occurs because is an odd function when n is odd, and thus its expectation is zero. When a pdf is less symmetric about its mean, odd central moments increase in value (except for the first central moment which is always zero). Consider the third-central moment:

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Figure 5.18 shows examples of for exponential and Rayleigh pdfs, both of which have unit variance so that is the same as skewness. We have also plotted the cumulative area

FIGURE 5.18 Functional view of skewness, showing FX(x), , and C(x). (a) Exponential with (). (b) Rayleigh with ().

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