Find the density function of $Z=XY$.
I was given $f(x,y)$ for $x>0$, $y>0$.
I know $F_{z}(z)= \iint f(x,y)dxdy$. But I need help finding the bounds for the double integral. I think the first one goes from $0$ to infinity and the second one goes from $0$ to z, but I am unsure.
Thank you!
$\endgroup$ 31 Answer
$\begingroup$We have
$$P(XY<Z)=P(X<\frac{Z}{Y})=F(\frac{Z}{Y})$$
Now you simply differentiate
$$\frac{d}{dz}F=\int_0^{\infty}\frac{d}{dz}\int_0^{\frac{Z}{Y}}f(x,y)dxdy$$
By Leibnitz integral rule we have $$\frac{d}{dZ}F=\int_0^{\infty}\frac{f(\frac{Z}{y},y)}{y}dy$$
Where $$\frac{d}{dZ}F=p_z(z)$$
Note that the absolute value in the term $\frac{1}{y}$ is not required, as our domain is $x>0,y>0$.
$\endgroup$ 1
