Geoscience Reference
In-Depth Information
andsoforth.
Stationarity (or the spatial equivalent, homogeneity) can greatly simplify statisti-
cal analyses. As we saw in
Part II
, real situations can often be taken as approximately
stationary (or homogeneous), and so in practice, and with justification from the
ergodic hypothesis, time or space averages are typically used instead of ensemble
averages. Here we will assume that all three give the same value, and continue to
use ensemble averages.
13.3 Examples of probability densities
13.3.1 The Gaussian
The probability density of a Gaussian random variable
u(t)
is
1
√
2
πσ
e
−
(u
−
u)
2
/
2
σ
2
.
β(u)
=
(13.26)
This i
s comple
tely determined by two parameters, the mean
u
and the variance
σ
2
u)
2
. Central moments
(moments about the mean) depend only on
σ
2
:
=
(u
−
⎧
⎨
⎩
0
n
odd,
σ
n
n
!
2
n/
2
(n/
2
)
!
(u
−
u)
n
=
(13.27)
n
even.
The skewness of a Gaussian random variable is 0 and the flatness factor is 3.
Figure 13.2
shows the Gaussian probability density and probability distribution
functions. The distribution function involves the
error function
(erf).
The Central Limit Theorem says that the probability distribution of the sum of
statistically independent random variables of an arbitrary but identical probability
distribution approaches a Gaussian in the limit as the number of elements in the sum
becomes large. In particular, the normalized sums of a large number of independent
events - for example, the errorsmade in performing an experiment - have aGaussian
distribution.
If
u(t)
and
v(t)
are jointly Gaussian, their joint probability density is
−
ρ
2
exp
u
2
σ
u
−
,
2
ρ
u
v
v
2
σ
v
1
2
πσ
u
σ
v
1
1
β(u,v)
=
−
σ
u
σ
v
+
2
(
1
−
ρ
2
)
(13.28)
where
u
v
σ
u
σ
v
,σ
u
=
u
=
v
=
(u
2
)
1
/
2
,σ
v
=
(v
2
)
1
/
2
.
u
−
u,
v
−
v,
ρ
=
(13.29)
