Reliable Computing 9: 91–108, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Statool: A Tool for Distribution Envelope
Determination (DEnv), an Interval-Based
Algorithm for Arithmetic on Random Variables
DANIEL BERLEANT, LIZHI XIE, and JIANZHONG ZHANG
Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011,
USA, e-mail: email@example.com
(Received: 1 July 2002; accepted: 8 October 2002)
Abstract. We present Statool, a software tool for obtaining bounds on the distributions of sums,
products, and various other functions of random variables where the dependency relationship of
the random variables need not be speciﬁed. Statool implements the DEnv algorithm, which we
have described previously  but not implemented. Our earlier tool addressed only the much more
elementary case of independent random variables . An existing tool, RiskCalc , also addresses
the case of unknown dependency using a different algorithm  based on copulas , while
descriptions and implementations of still other algorithms for similar problems will be reported soon
 as the area proceeds through a phase of rapid development.
The problem of determining derived distributions, the distributions of random vari-
ables whose samples are a function of samples of other random variables, has
received considerable attention. Springer’s monograph  is fairly comprehen-
sive up to its time of publication. Much of the subsequent work has focused on
copulas , which have motivated a number of conferences , , , .
Much of the existing work addresses analytical techniques. A shortcoming of the
analytical approach is its tendency to produce results applying to speciﬁc classes
of distributions, such as normal, lognormal, etc. Numerical methods form an alter-
native approach that tends to be applicable to a wider class of distributions. Monte
Carlo is the classical numerical method, but has some serious shortcomings ,
such as difﬁculties in safely handling problems in which dependency relationships
among random variables are unknown, or in which distributions are not fully spec-
iﬁed. Non-Monte Carlo numerical methods rely on discretization of distributions
followed by computation on the discretized forms.
Algorithms for non-Monte Carlo, numerical computation of derived distribu-
tions have been known since at least as early as 1968 . Early algorithms ,
, ,  assume that distributions whose samples are summed, multiplied,
etc., to yield the derived distribution of interest are independent. Furthermore they
rely on discretizations that approximate, as discretizations often do, implying that