Fundamentals of Probability and Statistics for Engineers

An important relation between the variance and simple moments is

This can be shown by making use of Equations (4.3). We get

fX(x)

Mode

Median

(a) Mean

fX(x)

Mode

Median

Mean

(b)

fX(x)

Mode

Median

(c) Mean

x

x

x

Figure 4.1 Relative positions of the mean, median, and mode for three distributions:

(a) positively shewed; (b) symmetrical; and (c) negatively shewed

80 Fundamentals of Probability and Statistics for Engineers

^2 ˆ 2 m^2 : … 4 : 8 †

^2 ˆEf…Xm†^2 gˆEfX^2  2 mX‡m^2 gˆEfX^2 g 2 mEfXg‡m^2

ˆ 2  2 m^2 ‡m^2 ˆ 2 m^2 :