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 .
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.