Calculate the product distribution

Question

Original question was to calculate the $X$, $Y$ and $XY$ distribution, however I have already done the first two, so I am just going to post the question the way I have it. $$ f_{X,Y}(x,y) = \left\{ \begin{array}{ll} ye^{-xy-y} & \textrm{when $x>0,y>0$}\\ 0 & \textrm{in other case}\\ \end{array} \right. $$ I calculated both $$ f_X(x) = \left\{ \begin{array}{ll} \frac{1}{(x+1)^{2}} & \textrm{when $x>0$}\\ 0 & \textrm{in other case}\\ \end{array} \right. $$ $$ f_Y(y) = \left\{ \begin{array}{ll} e^{-y} & \textrm{when $y>0$}\\ 0 & \textrm{in other case}\\ \end{array} \right. $$ Since x and y are dependent, i think i can not use the formula given here https://en.wikipedia.org/wiki/Product_distribution, can someone help?

Answer 1

If we assume that $XY=Z$, then $X=Z/Y$, and $f_Z(z)=0,\ \forall z\le0$.

For $z>0$: $$\begin{align}f_Z(z)&=\int_{-\infty}^{\infty}f_{X,Y}(z/y,y)\frac{1}{|y|}dy\\ &=\int_{0}^{\infty}e^{-z-y}dy\\ &=e^{-z} \end{align}$$