Geoscience Reference
In-Depth Information
Since from the definition of the probability distribution we can write
P(u
2
;
t)
−
P(u
1
;
t)
=
Pr
{
u
1
≤
u(t) < u
2
}
P(u
2
;
−
P(u
1
;
t)
t)
u
1
)
=
=
(u
2
−
β(u
1
;
t)u,
(13.7)
u
2
−
u
1
for
u
t)u
as the probability that values of
u(t)
occur
in a narrow band of width
u
lying above
u
. Thus
β
has the properties
→
0 we can interpret
β(u
;
+∞
β(u
;
t)
≥
0
,
β(u
;
t)du
=
1
.
(13.8)
−∞
13.2.2 Moments and characteristic functions
The probability density
β(u
;
t)
yields the
moments
of
u(t)
- the mean values of
powers of
u(t)
:
+∞
u
n
β(u
u
n
(t)
=
;
t)du.
(13.9)
−∞
This multiplies each value of
u
n
by the prob
ab
ility of occurren
c
e of that value and
sums the products. In the general case when
u
=
0using
(u
−
u)
n
in the integrand
yields
central moments
,or
moments about the mean
(Problem 13.15)
.
If the probability density is symmetric about the origin, all odd moments are zero.
Thus odd moments, suitably nondimensionalized, are measures of the asymmetry
of the probability density. The thirdmoment, nondimensionalizedwith the variance,
is called the
skewness
:
u
3
(t)
(u
2
(t))
3
/
2
.
S
=
(13.10)
The second moment is called the
variance
; the fourth moment, when made
dimensionless with the square of the second, is called the
kurtosis
†
or
flatness
factor F
:
u
4
(t)
(u
2
(t))
2
.
F
=
(13.11)
It describes the shape of the probability density. For a Gaussian distribution
F
=3.
The Fourier transform of
β(u
;
t)
is called the characteristic function
f(κ
;
t)
:
+∞
e
iκu
β(u
;
t)du
=
f(κ
;
t),
−∞
+∞
(13.12)
1
2
π
e
−
iκu
f(κ
β(u
;
t)
=
;
t)dκ.
−∞
†
A function with a peaked probability density is sometimes referred to as
kurtic
.
