PDF of dependent variables In my recent question an answer was given, and I am able to compute it myself. Still, I'd like to understand where does that answer come from. Hence, what's the approach to handle dependent variables in order to get a PDF? Specifically, $X,Y$ are iid uniform[0,1] RVs, and $W$'s PDF is $f_W(w)=-\frac{1}{4}\ln\frac{w}{4}$, $w\in(0,1]$. I want (preferably, using pen and paper) to calculate the PDF of $Z=X+Y+\sqrt{(X-Y)^2+W}$. The PDFs of $A=X+Y$ and $B=\sqrt{(X-Y)^2+W}$ (separately) are known. How do I find the PDF of $Z$?
As the answer obtained via Mathematica/mathStatica is (piecewise) given by elementary functions, I'd expect $Z$'s PDF to be possible to calculate analytically (the software does it somehow, so a human should be able to do it as well). I browsed the web in search for theory of dependent RVs or anything related, but couldn't find anything useful (i.e., anything I'd be able to comprehend). Any help in providing hints, sketching the solution or specifically pointing to any relevant source (publication, textbook, etc.) will be appreciated.
EDIT Is it possible to find conditional probability $Pr(A|B)$ (or $Pr(B|A)$), given the (marginal) distributions of $A$ and $B$? Then the joint probability would be $Pr(A,B)=Pr(A|B)\cdot Pr(B)$. Or not?
This can be a general question if there exists a general answer, i.e., given $f_X(x)$, $f_Y(y)$ and $f_W(w)$, how to find $f_Z(z)$ when $Z=G(X,Y,g(X,Y,W))$.
If there's an approach that exploits particular forms of the functions $G$ and $g$ (from the first paragraphs of this question), I'd be happy to know it.
 A: I specify my approach:
Let $X=A+D+\sqrt{(A-D)^2+U}$ with $Y=A+D$ and $Z=(A-D)^2$; hence $A=\frac{Y+\sqrt{Z}}{2}$, $D=\frac{Y-\sqrt{Z}}{2}$ and $U=(X-Y)^2-Z$, with a Jacobian $|J|=\left|\frac{y-x}{2\sqrt{z}}\right|$. Then, the PDF of $X$ is
$$f_X(x)=-\frac{1}{4}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}f_A\left(\frac{y+\sqrt{z}}{2}\right)\cdot f_D\left(\frac{y-\sqrt{z}}{2}\right)\cdot\ln\frac{(x-y)^2-z}{4}\cdot\left|\frac{y-x}{2\sqrt{z}}\right|dydz,$$
where $f_A$ and $f_D$ are equal to 1 if their argument is in $[0,1]$ and equals 0 otherwise, and $-\frac{1}{4}\ln\frac{*}{4}$ is the PDF of $U$, with $*\in(0,4]$. The integration region is given by the inequalities
$$0\leq y+\sqrt{z}\leq 2$$
$$0\leq y-\sqrt{z}\leq 2$$
$$0<(x-y)^2-z\leq 4$$
with $x\in[0,4]$, $y\in[0,2]$ and $z\in[0,1]$. The integration region is additionaly split by $x=y$, as at this surface the Jacobian changes its sign and in the formula for $f_X(x)$ we have its module.
Can someone verify whether my reasoning and the formula for $f_X(x)$ are correct? And the last sentence about splitting the integral by the Jacobian.
(Yes, calculating this integral seems to be a pain; but is it at least correct?)
