Suppose we're given a vector of two random values $(\xi, \eta)$ uniformly distributed within a circle of constant radius (let it be 1). Does $\xi$ and $\eta$ are independent?
What I did:
Distribution functions of these values are similar and look like $\mathit{F}_\xi (x) = \begin{cases} 0, x\notin [-1,1]\\ \pm \sqrt{1-x^2}, x\in [-1,1]\end{cases}$.
Also their correlation is $0$.
Are they independent? I think they are and I know that correlation = $0$ can't really define if they are independent or not. But it seems like they have distribution functions which depend only from an area they are calculated in.