Means and Variances of Stochastic Vector Products with Applications to Random Linear Models

Abstract

Applications in operations research often employ models which contain linear functions. These linear functions may have some components (coefficients and variables) which are random. (For instance, linear functions in mathematical programming often represent models of processes which exhibit randomness in resource availability, consumption rates, and activity levels.) Even when the linearity assumptions of these models is unquestioned, the effects of the randomness in the functions is of concern. Methods to accommodate, or at least estimate for a linear function the implications of randomness in its components typically make several simplifying assumptions. Unfortunately, when components are known to be random in a general, multivariate dependent fashion, concise specification of the randomness exhibited by the linear function is, at best, extremely complicated, usually requiring severe, unrealistic restrictions on the density functions of the random components. Frequent stipulations include assertion of normality, or of independence-yet, observed data, accepted collateral theory and common sense may dictate that a symmetric distribution with infinite domain limits is inappropriate, or that a dependent structure is definitely present. (For example, random resource levels may be highly correlated due to economic conditions, and non-negative for physical reasons.) Often, an investigation is performed by discretizing the random components at point quantile levels, or by replacing the random components by their means-methods which give a deterministic "equivalent" model with constant terms, but possibly very misleading results. Outright simulation can be used, but requires considerable time investment for setup and debugging (especially for generation of dependent sequences of pseudorandom variates) and gives results with high parametric specificity and computation cost. This paper shows how to use elementary methods to estimate the mean and variance of a linear function with arbitrary multivariate randomness in its components. Expressions are given for the mean and variance and are used to make Tchebycheff-type probability statements which can accommodate and exploit stochastic dependence. Simple estimation examples are given which lead to illustrative applications with (dependent-) stochastic programming models.