Geoscience Reference
In-Depth Information
and that about the mean
x
=
μ
,as
+∞
)
n
m
n
=
(
x
−
μ
f
(
x
)
dx
(13.10)
−∞
Moments about the mean are also called central moments. In principle, the zeroth
moments equal unity, or
m
0
=
m
0
=
1. The first moment about the origin is the
mean
,
by definition, or
m
1
=
μ
; this is also called the
expected value
of the random variable,
E
{
X
}
. From Equation (13.10) with
n
=
1, it follows that the first moment about the mean
is zero, or
m
1
=
0. The second moment about the origin is called the
mean square devi-
ation
. The second moment about the mean is called the
variance
, and is usually denoted
by
m
2
=
σ
2
; its square root
σ
is called the standard deviation. When the standard devi-
ation is made dimensionless with the mean, it is called the
coefficient of variation
,or
C
v
=
). These parameters related to the second moment can be used to characterize
the dispersion of the random variable. All odd central moments of distributions with
a symmetrical distribution function are zero, or
m
3
=
(
σ/μ
0. The lack of sym-
metry or skew is commonly expressed by the third moment about the mean,
m
3
. The
coefficient of skew is defined by
C
s
=
m
5
= ···=
3
). A distribution function is symmetrical
(
m
3
/σ
about the origin
x
=
0, if
F
(
x
)
=
1
−
F
(
−
x
)
(13.11)
so that also
x
). The central moments can be readily obtained from the
moments about the origin. For the second and third central moments the relationships
can be shown to be
f
(
x
)
=
f
(
−
m
1
2
m
2
−
m
2
=
(13.12)
2
m
1
3
m
3
−
3
m
1
m
2
+
m
3
=
The same relationships also hold when
m
1
,
m
2
and
m
3
denote the moments about any
arbitrary reference, say
x
a
.
The moments of a distribution can be estimated directly from a set of
n
observa-
tions,
X
i
, with
i
=
2
, and skew
=
1
,
···
,
n
. Sample estimators of the mean
μ
, variance
σ
coefficient
C
s
, are, respectively,
n
X
i
i
=
1
M
=
X
i
=
n
n
M
)
2
(
X
i
−
(13.13)
i
=
1
S
2
=
n
−
1
n
M
)
3
n
(
X
i
−
i
=
1
g
s
=
2)
S
3
(
n
−
1)(
n
−
