Let μn denote the nth central moment of a random variable X. Two quantities of interest, in addition to the mean and variance, are

The value α3 is called the skewness and ou is called the kurtosis. The skewness measures the lack of symmetry in the pdf (see Exercise 2.26). The kurtosis, although harder to interpret, measures the peakedness or flatness of the pdf.€ƒ
(a) Show that if a pdf is symmetric about a point a, then a3 = 0.
(b) Calculate a3 for f(x) = e_x, x > 0, a pdf that is skewed to the right.
(c) Calculate cu for each of the following pdfs and comment on the peakedness -of each.

Ruppert (1987) uses influence Junctions (see Miscellanea 10.6.4) to explore further the meaning of kurtosis, and Groeneveld (1991) uses them to explore skewness; see also Balanda and MacGillivray (1988) for more on the interpretation of a4.

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43 (μ2)3/2 f(z) #-, -1 < z < 1 2 f(z) = 1e-lat z < oo -oo <