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Say there are random variables such that $X=Y+Z$ with $Y$, $Z$ independent; knowing the pdfs of $Y$ and $Z$, one can (technically) find the pdf of $X$. Taking it from the other side: if one knows the …

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\s …

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 …

I have four independent uniformly distributed variables $a,b,c,d$, each in $[0,1]$. I want to calculate the distribution of $(a-d)^2+4bc$. I computed the distribution of $u_2=4bc$ to be $$f_2(u_2)=-\ …

This is a direct continuation of my recent question. The thing that I actually want to get is the distribution of $a+d+\sqrt{(a-d)^2+4bc}$, where $a,b,c,d$ are uniform in $[0,1]$. Now, the distributio …

Just after reading wolfies answer I understood I could calculate the final distribution from the very beginning without all the mid-point steps: M[x_] := M[x] = Evaluate@FullSimplify@ Integrat …