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information on the
p
X
is needed. In many cases however, the available information is limited to
the mean and variance of
X
. Furthermore, even if
p
X
is known, the computation of the integrals
in Equations (3.2) and (3.3) may be time-consuming (Ang and Tang, 1975). Consequent-
ly, faster approximation methods are often preferred, that allow approximate values of the
mean and variance to be computed. The FOSM method is one of these approximate methods.
For the sake of generality, consider a function of several random variables
X
1
,…,
X
n
:
Y
=
y
(
X
1
,…,
X
n
)
(3.4)
Expanding the function in a Taylor series about the mean values
yields the
following expressions (see, e.g., Ang and Tang, 1975)
(3.5)
(3.6)
where
Cov (X
i
, X
j
)
is the covariance between
X
i
and
X
j
,
defined as
(3.7)
All derivatives are evaluated at the mean values The quantity #
y
/#
X
i
is called the
sensitivity of
Y
to the input variable
X
i
. The first term on the right-hand side of Equation
(3.6) represents the contribution of the variances of the input variables to the total
variance of the output. The second term denotes the influence of a possible correlation
among the various possible pairs of input variables. If the input variables are statistically
independent, i.e.
Cov(X
i
, X
j
)
=0, this second term vanishes and the variance of
Y
becomes
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