Characterizations of the Beta Distributions via Some Regression Assumptions

Let X and Y be two independent non-degenerate random variables. Also let (U, V ) be a bijective map of (X, Y ). It is desired to use certain regression assumptions between U and V to characterize the distributions of X and Y , and consequently, the distribution of (U, V ). In most of the previous investigations, U and V turn out to be independent too. Recently, for X, Y valued in (0, 1), Seshadri and Weso lowski (2003) charac- terize X and Y to be beta distributed based on two constancy of regression assumptions between U and V , where (U, V ) is a particular bijective map of (X, Y ). In this work, first we will generalize the results in Seshadri and Weso lowski (2003). It will be proved that for the bijective map given in Seshadri and Weso lowski (2003), X and Y are beta distributed under some more general regression assumptions. Next we illustrate that for some other special bijec- tive maps (U, V ), under certain regression assumptions between U and V , X and Y can also be characterized to be beta distributed, yet U and V may not be independent.