Calculate the product distribution 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. 
$$
f_{X,Y}(x,y) = \left\{ \begin{array}{ll}
ye^{-xy-y} &amp; \textrm{when $x&gt;0,y&gt;0$}\\
0 &amp; \textrm{in other case}\\
\end{array} \right.
$$
I calculated both
$$
f_X(x) = \left\{ \begin{array}{ll}
\frac{1}{(x+1)^{2}} &amp; \textrm{when $x&gt;0$}\\
0 &amp; \textrm{in other case}\\
\end{array} \right.
$$
$$
f_Y(y) = \left\{ \begin{array}{ll}
e^{-y} &amp; \textrm{when $y&gt;0$}\\
0 &amp; \textrm{in other case}\\
\end{array} \right.
$$
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?
If we assume that $XY=Z$, then $X=Z/Y$, and $f_Z(z)=0,\ \forall z\le0$. 

For $z&gt;0$: $$\begin{align}f_Z(z)&amp;=\int_{-\infty}^{\infty}f_{X,Y}(z/y,y)\frac{1}{|y|}dy\\
&amp;=\int_{0}^{\infty}e^{-z-y}dy\\
&amp;=e^{-z}
\end{align}$$
If we assume that $XY=Z$, then $X=Z/Y$, and $f_Z(z)=0,\ \forall z\le0$. 

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