This example deals with 48 uncertain parameters, which are considered as uniform

random variables having an uncertainty of 20% around their nominal values given in Fig.

Probabilistic load flow, which incorporates dependence between

random variables, is an efficient tool in probabilistic analysis as it enables a comprehensive assessment of system working conditions, thus could inform system operators of the weak points and potential crisis under various uncertainties [5].

From Theorem 1, the extended Birnbaum-Saunders distribution has density (9) and from Theorem 3 we can generate

random variables X ~ EBS([alpha], [beta], [xi]).

Denuit, "Comonotonicity, orthant convex order and sums of

random variables," Statistics & Probability Letters, vol.

Based on the superposition principle, the random differential equation with an input depending on several

random variables is decomposed on a sequence of RDE with the same main random operator and reduced right-hand sides.

Let [([F.sub.n]).sub.n[member of]N] be a filtration and [([X.sub.n]).sub.n[member of]N] be a sequence of

random variables. We say that [([X.sub.n]).sub.n[member of]N] is a martingale adapted to the filtration [([F.sub.n]).sub.n[member of]N] if for every n [member of] N

Since the states (the values of the

random variable [X.sub.k]) are the same for each k, one only needs the second row to describe the pdf.

where [y.sup.(i)] is the ith realization of

random variable Y and n is the number of available sample points.

Definition 4:

Random variable y is larger than x in the increasing convex order if [E.sub.G][u(x)] [greater than or equal to] [E.sub.F][u(x)] for all u(x) with u' (x) [greater than or equal to] 0 and u"(x) [greater than or equal to] 0.

Apparently, Haavelmo was simply 'considering' that economic variables are

random variables because he needed this assumption.

In structural reliability problem, let us assume that the limit-state function is g(x, y) = g([x.sub.1], [x.sub.2], ..., [x.sub.n], [y.sub.1], [y.sub.2], ..., [y.sub.m]), where x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) is a n-dimensional vector of

random variables and y = ([y.sub.1], [y.sub.2], ..., [y.sub.m]) is a m- dimensional vector of interval variables.

Figure 2 demonstrates how the hyperplane (line), which is the line of a constant sum of the values of the

random variables and is perpendicular to the n-cube's (square's) main diagonal, accrues volume (area) below it.