Suppose two non-negative random variates x and y are NOT independent (in my case that I am interested in, the range of x is constrained by y, i.e., $0<x\leq y$).
In more specific, assuming $y$ follows a uniform distribution $U(0,100)$ while $x$ is always $\leq y$, so now is $z=x/y$ independent of y? why? can you provide a proof or reference for this?
My idea is that probability distribution function for z should be derived. here z has a range $0<z\leq1$.
For testing independence of y and z, we should prove that $f(z,y)=f(z) \cdot f(y)$. But I did not have a clue on it.