Let $(X,Y)$ be a continuous 2-dimensional random variable with joint density $f_{X,Y}(x,y)$. I want to derive the formula for $f_z(z)$, that is, the density function of $Z=X+Y$.
Let $\Omega=$$\{x,y : x+y<z\}$. Then:
$F(z):=P(Z<z)=P(X+Y<z)=\int \int_{\Omega}f_{X,Y}(x,y)dxdy = \int_{-\infty}^{\infty} \int_{-\infty}^{z-x}f_{X,Y}(x,y)dxdy$
Now, to get $f_z(z)$, we just need to evaluate $\frac{d}{dz}F(z)$. My problem is that i have no idea, from here on, how to get that:
$f_z(z)=\int_{-\infty}^{\infty}f_{X,Y}(x,z-x)dx$
I have tried change of variables with no success, still stuck on a double integral instead of a single one. Any ideas?
Haven't found anything on the internet either, so a link for the derivation would be helpful aswell. Thanks
\Edit: I'm only interested with the derivation starting with that approach. I'm familiar with the transformation theorem and how to derive this formula using it.