This technique represents the random response of a model in terms of a linear combination of orthonormal polynomial basis involving the input random variables
. Intrusive [3-5] or non-intrusive [2, 6, 7] approaches exist for the computation of the basis coefficients, thus reproducing an analytic representation called metamodel (or surrogate model) of the system response.
The vth moment of a random variable
X, whose PDF f (X) is given, can be defined as
(3) The r-th moment of the random variable
X is given by
Let [l.sub.i] and [[mu].sub.i] be the essential infimum and essential supremum of a random variable
[X.sub.i], X = ([X.sub.1],..., [X.sub.n]) is a fixed random vector in [mathematical expression not reproducible], then X is mutually exclusive.
For h [member of] N and i between 0 and n, the moment [m.sub.ih] of order h of the random variable
[[xi].sub.i] is defined by
Let X : [OMEGA] [right arrow] R be a random variable
with finite expected value and let H [subset or equal to] S be a [sigma]-algebra.
If X(S) [??] R is at most countable in R, then X is a discrete random variable
, otherwise, it is a continuous random variable
Damage is a random variable
because it is different for each trajectory.
There is no agreed-upon measure of the size of a random variable
, nor is there an agreed-upon measure of the magnitude of risk.
However, this definition has never been used as the criterion for the assumption of a random variable
in a macroeconometric model.
And G (u, [delta]) can be transformed to G(v) = 1 - [square root of ((2/(n + m)))] [v.sub.1][v.sub.2] in polar space, where the PDF of the random variable
[v.sub.1] is (18), and the PDF of [v.sub.2] is (20).
The present goal is to derive the CDF and the PDF of the sum T = [[summation].sup.n.sub.i=1] [X.sub.i], where [X.sub.i] are independent identically distributed Uniform (0, 1) random variables
for i = 1, 2, ..., n.