The joint density of X and Y is given by f (x,y) = xe^(-x(y+1)); x>0,y>0 0; otherwise

\[x e^{-x(y+1)}\]

Find the distribution of Z=XY using distribution function method

@Zarkon

@JamesJ I know you're doing ajp's problem, but I wanna know how to do this one as well if you get the chance

Just compute \[P(Z<z)=P(XY<z)\] integrate over the region in which \(XY<z\) is true

How to integrate it when it has two variables?

x y < z x < z/y You should compute the integral below to F(z) the CDF and F'(z) would be the pdf \[ \int _0^{\infty }\int _0^{\frac{z}{y}}x e^{-x y-x}dxdy \] Do you know how to compute this integral?

you compute first the integral below by parts \[ \int_0^{\frac{z}{y}} x e^{-x y-x} \, dx= \frac{1-e^{-\frac{(y+1) z}{y}} \left(\frac{z}{y}+z+1\right) }{(y+1)^2} \] It is messy, who said life is easy.

Then you integrate \[ \int_0^{\infty } \frac{1-e^{-\frac{(y+1) z}{y}} \left(\frac{z}{y}+z+1\right) }{(y+1)^2} \, dy= 1- e^{-z} \] I have to confess that I did not do the above integral by hand and I refuse to do it by hand.

So \[ F(z) = 1 - e^{-z}\\ f(z)=F'(z) = e^{-z} \]

Thank you for your big help. I will try @eliassaab :)