Page 97 - Fundamentals of Probability and Statistics for Engineers

80 Fundamentals of Probability and Statistics for Engineers

f (x)
X

x
(a) Mode Mean
Median
f (x)
X

x
(b) Mode
Median
Mean
f (x)
X

x
(c) Mean Mode
Median
Figure 4.1 Relative positions of the mean, median, and mode for three distributions:
(a) positively shewed; (b) symmetrical; and (c) negatively shewed

An important relation between the variance and simple moments is
2
2
ˆ 2 m : 4:8†
This can be shown by making use of Equations (4.3). We get
2
2
2
2
2
ˆ Ef X m† gˆ EfX 2mX ‡ m gˆ EfX g 2mEfXg‡ m 2
2
2
2
ˆ 2 2m ‡ m ˆ 2 m :

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