I read on here that given two jointly distributed random variables $X$ and $Y$, the density of their sum (let $Z=X+Y$) can expressed as
$$f_Z(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{z-x} f_{X,Y}(x,y)dydx \,.$$
I am trying to figure out where this formula comes from. If $$ \int_{-\infty}^{\infty}\int_{-\infty}^{z-x} f_{X,Y}(x,y)dydx = \int_{-\infty}^{\infty} f_{X,Y}(x,z-x)dx \,, $$ then I see how the original equation represents the convolution of $f_X$ and $f_Y$, which results in the density of $Z$. However, I am not sure if the expressions above are actually equal. Could someone please help me understand how this formula is derived?
The double integral that you've written is the CDF of $Z$, not the PDF. Indeed, by the fundamental theorem of calculus, $$\frac{d}{dz} \int_{-\infty}^\infty \int_{-\infty}^{z-x} f_{X,Y}(x,y) \, dy \, dx = \int_{-\infty}^\infty f_{X,Y}(x, z-x) \, dx$$ (assuming you can push the derivative past the first integral).