Functional equations stemming from probability theory

Abstract

Special cases of the functional equation
\[
h_{1}\left(\frac{x}{c\left(y\right)}\right)\frac{1}{c\left(y\right)}f_{Y}\left(y\right)=
h_{2}\left(\frac{y}{d\left(x\right)}\right)\frac{1}{d\left(x\right)}f_{X}\left(x\right)
\]
are investigated for almost all

$\left(x,y\right)\in\r^{2}_{+}$,
for the given functions

$c$, $d$

and the unknown functions

$h_{1}$, $h_{2}$,
$f_{X}$ and $f_{Y}$.