Reliable Computing 9: 127–141, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Estimating and Validating the Cumulative
Distribution of a Function of Random Variables:
Toward the Development of Distribution
WELDON A. LODWICK
Department of Mathematics, Campus Box 170, University of Colorado, P.O. Box 173364, Denver,
CO 80217–3364, USA, e-mail: Weldon.Lodwick@cudenver.edu
K. DAVID JAMISON
Watson Wyatt & Company, 950 17th Street, Suite 1400, Denver, CO 80202, USA,
(Received: 24 June 2002; accepted: 13 November 2002)
Abstract. A method for estimating and validating the cumulative distribution of a function of random
variables (independent or dependent) is presented and examined. The method creates a sequence
of bounds that will converge to the distribution function in the limit for functions of independent
random variables or of random variables of known dependencies. Moreover, an approximation is
constructed from and contained in these bounds. Preliminary numerical experiments indicate that this
approximation is close to the actual distribution after a few iterations. Several examples are given to
illustrate the method.
A method for estimating the cumulative distribution function (c.d.f.) for a real-
valued function of a ﬁnite set of random variables is described. Several illustrations
of the method are presented. When the number of random variables is moderate,
the method gives good results quickly. The estimate is a closed form approximation
that might prove useful in stochastic and fuzzy/possibilistic programming problems
(see  and  for early applications to optimization) and perhaps as an alternative
to simulations when closed forms are needed. We note the similarity between our
method with that of R. E. Moore  and several others before and after Moore
(see , , , , , , , , , a discussion as to the similarities
and differences between these methods and the one we develop here is given in the
next section). The method is an extension and application of the ideas discussed
in  and . In essence, the c.d.f. is bounded by consistent possibility and
necessity distributions (consistent with the underlying probability distribution), as
will be explained below.