Evaluation of minimum and maximum of a correlated pair of random variables via the bivariate laguerre transform

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Abstract
In a recent paper by Sumita and Kijima (1985), the bivariate Laguerre transform has been developed for mechanizing various bivariate continuum operations such as multiple bivariate convolutions, marginal convolutions, tail integration, partial differentiation and multiplication by bivariate polynomials. The transform has been shown to be quite useful and powerful in studying bivariate distributions and bivariate stochastic processes. In this paper, we study the minimum and maximum of a correlated pair of random variables via the bivariate Laguerre transform, thereby further enhancing the applicability of the bivariate Laguerre transform. A numerical procedure is developed for calculating the Laguerre coefficients of the probability density functions for the minimum and maximum in terms of the original bivariate Laguerre coefficients corresponding to the correlated pair of random variables. This enables one to evaluate the distributions and moments of the minimum and maximum. The joint distribution of the minimum and maximum can be also computed. Furthermore, time dependent behavior of stochastic models where the underlying lifetime distributions involve such minimum and maximum can be analyzed. Numerical examples are given, demonstrating speed and accuracy of the procedure developed.