Suppose $V,W,$ and $X$ are mutually independent random variables. Further let $Y=V+W$ and $Z=V+X$.
Is there a way to characterize the joint density $f_{Y,Z}(y,z)$ given the dependence of $Y$ and $Z$? By convolution, I am able to characterize the marginals, that is, $f_Y(y)$ and $f_Z(z)$, but I fail to characterize the joint density function.
In my specific case, I am considering uniform random variables, so, $V \sim U[-1,1]$, $W \sim [- \epsilon, \epsilon]$, and $X \sim U[-1,1]$. Therefore, I would highly appreciate an answer for the general or the specific uniform case (if possible with some citation to enhance my knowledge).
Thank you!
The joint cdf of $Y$ an $Z$ is defined as
$$F_{Y,Z}(y,z)=$$$$=P(V+W<y\cap V+X<z)=$$$$=\frac12\int_{-1}^{1}P(v+W<y\cap v+X<z\mid V=v)dv.$$
Since our random variables are independent, we may say that
$$ P(v+W<y\cap v+Y<z\mid V=v)=P(W<y-v)P(X<z-v).$$
The probabilities above can be calculated and then the integral with respect to $z$ can be evaluated.
Finally, we will have to calculate the parcial derivatives of the integral.