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 V, 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 Wesolowski (2003) characterize 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 Wesolowski (2003). It will be proved that for the bijective map given in Seshadri and Wesolowski (2003), X and Y are beta distributed under some more general regression assumptions. Next we illustrate that for some other special bijective 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.