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(4.55)
Now, referring to Equation (4.49), if the function
f
is monotonically decreasing with
x,
then the CFD of
Z
is given by (Ang and Tang, 1975, p. 171):
P
Z
(z)
=1!
P
X
[
f
!
1
(z)
]
(4.56)
For a linear relationship between
X
and
Y
(i.e.
Y
=
a
+
bX
), with
b
!0, the function
f
becomes monotonically decreasing with
x
if
Z
=
X
!
Y
. Then from Equation (4.56), it can
be shown (for subtraction) that
(4.57)
Similarly, for division, i.e.
Z
=
X/Y,
with
z
monotonically decreasing with
x
and
y
%0, we
obtain:
(4.58)
It can be observed that Equations (4.43) and (4.45) are similar to Equations (4.54) and
(4.55), respectively. These two sets of equations are for addition and multiplication. This
suggests that the addition and multiplication of two functionally dependent random
variables (if their sum and product are monotonically increasing) can be made equivalent
to the corresponding operations on fuzzy variables, by applying an appropriate
transformation between their CDF and MF.
The operations are not similar in the case of subtraction and division. The difference is
due to the reversed order of the appearance of the lower and upper bound values of the
operating variables in fuzzy subtraction and division (Equations (4.44) and (4.46)).
4.4.3
Probability-fuzzy transformations
Let us define two transformations:
T
1
from probability (in the form of a CDF) to fuzzy
(in the form of a MF) and
T
2
from MF to CDF for two fuzzy operations (
)
[(+),($)] and
two random operations
[+,$]. The transformation
T
1
is such that
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