Multivariate Exponential and Geometric Distributions with Limited Memory

Abstract

Many connections between geometric and exponential distributions are known. Characterizations and derivations of these distributions often run parallel. Moreover, one kind of distribution can be derived from the other: exponential distributions are limits of sequences of rescaled geometric distributions, and the integral parts of exponential random variables have geometric distributions. The lack of memory property is expressed by functional equations of the form F ( x + u) = F ( x) F ( u) for all ( x, u) ∈ S , where F ( x) = P{ X 1 > x 1, ..., X n > x n }. With S = R 2 n +, the equation expresses a complete lack of memory that is possessed only by distributions with independent exponential marginals. But when S is a proper subset of R 2 n +, the functional equation expresses a partial lack of memory property that in some cases is possessed by a more interesting family of multivariate exponential distributions, as for example, those with exponential minima. In this paper appropriate choices for S and the resulting families of solutions are investigated, together with the associated families of multivariate geometric distributions.