563489578934

B–6 Ensemble Average and Moments 693
score of 100, 2 with scores of 95, 4 with 90, 6 with 85, 10 with 80, 10 with 75, 5 with 70, 1 with
65, and l paper with a score of 60. Then, the class average is
xq =
100(1) + 95(2) + 90(4) + 85(6) + 80(10) + 75(10) + 70(5) + 65(1) + 60(1)
= 100a 1
40 b + 95a 2 40 b + 90a 4
40 b + 85a 6 10
b + 80a
40 40 b + 75a10 40 b
+ 70a 5
40 b + 65a 1
40 b + 60a 1
40 b
40
= a
9
i = 1
x i P(x i ) = 79.6
(B–24)
Moments
Moments are defined as ensemble averages of some specific functions used for h(x). For
example, for the rth moment (defined subsequently), let y = h(x) = (x - x 0 ) r .
DEFINITION.
given by
The rth moment of the random variable x taken about the point x = x 0 is
(x - x 0 ) r q
= (x - x 0 ) r f(x) dx
L
-q
(B–25)
DEFINITION. The mean m is the first moment taken about the origin (i.e., x 0 = 0).
Thus,
q
m ! xq = xf(x) dx
L
-q
(B–26)
s 2
DEFINITION. The variance is the second moment taken about the mean. Thus,
q
s 2 = (x - xq) 2 = (x - xq) 2 f(x) dx
L
-q
(B–27)
DEFINITION.
The standard deviation s is the square root of the variance. Thus,
q
s = 2s 2 = (x - xq) 2 f(x) dx
C L
-q
(B–28)
As an engineer, you may recognize the integrals of Eqs. (B–26) and (B–27) as related to
applications in mechanical problems. The mean is equivalent to the center of gravity of a mass
that is distributed along a single dimension, where f(x) denotes the mass density as a function
of the x-axis. The variance is equivalent to the moment of inertia about the center of gravity.