Some Bivariate Distributions of (X, Y) Where the Conditional Distribution of Y, Given X, is either Beta or Unit-Gamma

Summary

Bivariate distributions of continuous, non-negative r.v.’s (X,Y), where Y is an additive component of X, occur in many practical problems, e.g. survival data analysis, damaged observations, weather studies, etc. In such problems the original r.v. X may not be observable. However, if the conditional distribution of Y, given X, is assumed, then some of the distributional properties of the r.v. X are tractable through the observed r.v. Y. In this paper, two such conditional distributions, namely, the beta and unit-gamma are considered and the corresponding bivariate distributions of (X,Y), when X has the lognormal, Pareto and generalized gamma distribution, respectively, are studied. Parameter estimation for these bivariate distributions is discussed. Interpretation and application of the quantities E[Y]/E[X] and V(log Y) are suggested. The effect of assuming the above mentioned conditional distributions on these quantities is stressed.