Is there a way to derive $X+Y$ when knowing only the joint distribution of $X$ and $Y$? For example, the joint density is $\frac{1}{4}$, on the interval $(0 \leq x \leq 2)$ and $(0 \leq y \leq 2)$? can we know the distribution of $X+Y$?
My intuition is to factor the joint distribution, get the marginal and do convolution, in this case is just two i.i.d $Unif(0,2)$ and that should be easy. However, I really want to know is there a way I can skip the marginal part of the calculation and do things directly on the joint p.d.f? please tell me! any help a appreciated.
The distribution function of $X+Y$ is given by $P(X+Y \leq z)=\int_{-\infty} ^{\infty} \int_{-\infty}^{ z-u} f(u,v)dvdu$ and the density of $X+Y$ is the derivative of this function. [Here $f$ is the joint density of $(X,Y)$].