Geoscience Reference
In-Depth Information
If
g
is a monotonically increasing function of
x
with a unique inverse
g
!1
, then the CDF
of
Y
can be related to the CDF of
X
by the relationship (Ang and Tang, 1975, p. 170):
P
Y
(y)
=
P
X
[
g
!1
(y)
]
(4.48)
where
g
!1
is the inverse function of
g
.
The summation
Z
of
X
and
Y,
which are related by the function
g,
is given by
Z
=
X
+
Y
=
X
+
g(X)
=
f (X)
(4.49)
Now, if the function
f
is also monotonically increasing function of
x,
then using Equation
(4.48), the CDF of
Z
can be expressed in terms of the CDF of
X
as follows:
P
Z
(z)
=
P
X
[
f
!
1
(z)
]
(4.50)
The simplest relationship between
X
and
Y
that keeps the function
f
monotonically
increasing with
x
is a linear relationship, i.e.
Y
=
a
+
bX,
with
b
!0.
From Equation (4.50), for any value
&
=[0, 1], it follows
(4.51)
and
(4.52)
Since
z
is related to
x
as given by Equation (4.49), we obtain
(4.53)
Now, from Equations (4.51) and (4.53), we obtain
(4.54)
Similarly, the following can be derived for multiplication between
X
and
Y
(
Z
=
X Y,
X
=
g(Y)
), if
Z
is monotonically increasing as follows:
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