Geoscience Reference
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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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