On generalized transforms of distribution functions

Abstract

In this note we discuss the following problem. LetX andY to be two real valued independent r.v.'s with d.f.'sF and ϕ. Consider the d.f.F*ϕ of the r.v.X oY, being o a binary operation among real numbers. We deal with the following equation: $$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$ where $$\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 $$ are real or complex functionals, т another binary operation ands a parameter. We give a solution, that under stronger assumptions (Aczel 1966), is the only one, of the problem. Such a solution is obtained in two steps. First of all we give a solution in the very special case in whichX andY are degenerate r.v.'s. Secondly we extend the result to the general case under the following additional assumption: $$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$ .