Bivariate distributions with Weibull conditionals (Q923553)

The authors consider the functional equation \[ (1)\quad f(x,y)=f(y| x)g(x)=f(x| y)h(y) \] where f(y\(| x)\), f(x\(| y)\) and g(x), h(y) are the conditional and marginal densities of a random vector (X,Y), respectively. The solution of (1) is obtained when the conditional family f(y\(| x)\) is assumed to be Weibullian, f(x\(| y)\) either Weibull or gamma and all the functions in (1) are twice differentiable. The suggested method of characterizing distributions through conditionals, as the authors say, is applicable to several other models as well.