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 [4] but not implemented. Our earlier tool addressed only the much more elementary case of independent random variables [3]. An existing tool, RiskCalc [13], also addresses the case of unknown dependency using a different algorithm [33] based on copulas [23], while descriptions and implementations of still other algorithms for similar problems will be reported soon [17] as the area proceeds through a phase of rapid development.

When random variables possessing arbitrary distribution functions must be combined via +, −, *, /, min(), max(), etc., Monte Carlo simulation is commonly employed. However, Monte Carlo simulation assumes either independence or (less commonly) some other specific dependency relationship, among other limitations (Ferson 1996). Discretization of the distribution function followed by a numerical method is an alternative. Numerical methods can relax the requirement of Monte Carlo that the distributions have a known dependency relationship, in which case the results are typically envelope curves within which the cumulative distribution of the result must lie regardless of the dependency relationship between the operands. The operands themselves can also be expressed with envelopes in order to bound the effects of discretization of the input distributions (Berleant 1993; Williamson and Downs 1990). This paper describes Statool, a software tool that implements Distribution Envelope Determination (DEnv), a numerical algorithm for performing arithmetic on distribution function operands (Berleant and Goodman-Strauss 1998). Our previously reported tool was limited to independent random variables (Berleant and Cheng 1998), a significant limitation. Improvements to Statool are currently being driven by the needs of applications in accordance with our research strategy, which is to identify such applications and then to modify Statool as needed to support them. However, identifying good applications is itself a research topic. We are currently exploring applications to the electric power industry (Sheblé and Berleant 2002; Berleant et al. 2002), and have obtained recent results on time to completion of multiple tasks and time to failure of two components [7,8].

When random variables possessing arbitrary distribution functions must be added, subtracted, multiplied, etc., Monte Carlo simulation is commonly employed. However, Monte Carlo simulation typically assumes that the distribution functions to be combined are independent, and must assume either independence or some other specific dependency relationship. Discretization of the distribution function followed by a numerical method is another alternative approach. Numerical methods can relax the requirement of Monte Carlo that the distributions must have a known dependency relationship by producing boundary curves within which the cumulative distribution resulting from any dependency relationship must lie. Furthermore, numerical methods can account for discretization error by producing boundary curves that are more inclusive than they would be if the same problem was run with a finer discretization, thereby accounting for discretization error in the results. This paper describes Statool, a software tool available in source and binary, that runs an interval-based numerical method for performing arithmetic operations on distribution functions that are either independent or have an unknown dependency relationship, and provides inclusion guarantees deriving from its interval based calculations.