Many mathematical models in operations research require computation of products of vectors whose elements are random variables. Unfortunately, analytic results for functions of interest are only obtained through highly restrictive, often unrealistic, choices of prior densities for the vectors' elements. Often, an investigation is performed by discretizing the random variables at point-quantile levels, or by outright simulation. This paper addresses the problem of characterizing the inner product of two stochastic vectors with arbitrary multivariate densities. Expressions for means of variances of vector products are obtained, and used to make Tchebycheff-type probability statements. Included are applications to stochastic programming models. (Author)