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 [15], the Expansion Optimal Linear Estimation method (EOLE) [13] or the Orthogonal Series Expansion method (OSE) [16] can characterize the process in the time domain.

Random variable 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 [9, 10].

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 or processes.