I have a question about my naive heuristical attempt for deriving the distribution $F_{XY}(x)$ for the product $Z=XY$ of tho independend random variables $X,Y$ with distributions $f_X(x),f_Y(x)$.
As for sum $X+Y$ we can understand $f_{X +Y}(z) = \int_{-\infty} ^{\infty} f_X(x)f_Y(z-x)dx$ as "summation" (=integration) over all real $x$ with $x + (z-x)=z$ for fixed $z$,
Where the same heuristic attempt for calculating $f_{XY}(z)$ fails?
Indeed, I know that $f_{XY}(z)= \int _{{-\infty }}^{{\infty }}f_{X}\left(x\right)f_{Y}\left(z/x\right){\frac {1}{|x|}}\,dx$,
also the wiki proof is also clear to me.
But where is my logical fail if I heuristically argue as for sum above by gathering (sum, resp. integrate) over all $x$ with $x*z/x=z$?
In this manner I would get the wrong formula $f_{XY}(z)= \int _{{-\infty }}^{{\infty }}f_{X}\left(x\right)f_{Y}\left(z/x\right)\,dx$