Let X be a random vector of joint PDF [f.sub.X] (x), including M random variables
([X.sub.1], ..., [X.sub.M]) assumed to be independent and representing the uncertain input parameters of the problem.
where [[mu]'.sub.1] = m is the mean of the random variable
and is used to calculate the central moment as follows
(3) The expected value and variance of the random variable
X is given by
Stoploss order as a special case of convex order is the most frequently used order relation for the comparison of risks, written as X [[less than or equal to].sub.sl] Y, for any two random variables
X and Y, if and only if the inequality E[[(X - d).sub.+]] [less than or equal to] E[(Y - d)+] holds for all real d, where [([xi]).sub.+] denotes the positive part of the real In addition, X is said to precede Y in the convex order sense; define X [[less than or equal to].sub.cx] 7, if and only if X [[less than or equal to].sub.sl] Y and E[X] = E[Y].
The second one, elaborated in this paper, consists of determining the 1D polynomial chaos associated with each random variable
[[xi].sub.k] [less than or equal to] k [less than or equal to] 2 and using the bijection [[psi].sub.2] to build the 3D polynomial chaos related to the random vector ([[xi].sub.0], [[xi].sub.1], [[xi].sub.2])).
(2) Let X : [OMEGA] [right arrow] R be a random variable
and [G.sub.1] [subset or equal to] [G.sub.2] [subset or equal to] S are [sigma]-algebras.
If X is a discrete random variable
, then a better way of describing it is to give its probability distribution function (pdf) or probability mass function (pmf), an array that contains all its values [x.sub.i], and the corresponding probabilities with which each value is taken, [p.sub.i] = P(X = [x.sub.i]),
Each trajectory is cycle counted and the Miner's model is used to estimate the corresponding realization of the damage random variable
. For a stationary and Gaussian process, the Karhunen Loeve (K-L) expansion [L3, 14], the Shinozuka method , the Expansion Optimal Linear Estimation method (EOLE)  or the Orthogonal Series Expansion method (OSE)  can characterize the process in the time domain.
y is said to be larger than another random variable
x in the increasing convex order if y is preferred or indifferent to x by all decision makers with increasing and convex utility functions.
Since 'uncertainty' is not a random variable
with certain distribution, there is no way macroeconometricians can model 'uncertainty' successfully.
In FORM for structural reliability analysis, by the first-order Taylor series, the limit-state function is approximated at the design point in the standard Gauss space of transformed independent random variables
Some derivations employ characteristic functions in a variety of ways, since the characteristic function of a sum of independent random variables
is the product of each summand's characteristic function and the inverse transform is not intractable ([11, pages 188-189], [12-14], [15, pages 362-363], [16, 17]).
where p represents the vector of design variables (five y coordinates and 14 member areas) and u represents the random variable
vector (14 member areas and the Young modulus).
Results obtained with clinical indicators TO, RAS, and p53 are compared to Adj-[R.sup.2] values obtained using random variable
for each indicator.
Actually, several factors such as the vehicles' parameters, the wind velocity, and the road surface roughness are random variables