I keep running into implicit assumptions re claims that continuous random variables are uncorrelated.
Specifically, I am reading that points on the circle, i.e., the random variables $X, Y= \pm \sqrt {1 - X^2}$ are uncorrelated, without specifying the joint of $X,Y$. We can only tell, AFAIK:
$$ E(XY) - \mu_X \mu_Y = \int xyf_{XY}(x,y) dxdy - \int xf_X(X) \int yf_Y(Y)dy =0 $$ EDIT: I only need to figure out $f_{xy}(x,y)$ and then I can find the marginals.
Is there a way of finding all joints $f_{xy} (x,y)$ that would solve above equation?