Finding a probability density function of a function of three dependent random variables

Question

I have three random variables that are functions of another three random variables by pairs, say: $U=fc(X,Y)$, $V=fc(Y,Z)$ and $W=fc(X,Z)$, with $X$, $Y$ and $Z$ being independent random variables with known pfds $f_X(x)$, $f_Y(y)$ and $f_Z(z)$, respectively. The densities $f_U(u)$, $f_V(v)$ and $f_W(w)$ are also known.

I would like to know the distribution $f_K(k)$, with $K=U+V+W$.

Clearly $U$, $V$ and $W$ are correlated and thus the usual variate transformation cannot be applied.

Any hints on how to tackle this problem?

Answer 1

In general, I would try finding the CDF. $$ F_K(k) = \iiint_{\{(x,y,z): f_c(x,y)+f_c(y,z) + f_c(x,z)\le k\}} f_X(x) f_Y(y) f_Z(z)\; dx\; dy\; dz$$