Suppose that the random vector $(X, Y)$ is uniformly distributed over the unit ball in $R^2$. Calculate $Cov(X, Y)$

Question

Suppose that the random vector $(X, Y)$ is uniformly distributed over the unit ball in $\mathbb R^2$. Calculate $Cov(X,Y)$

I'm not sure how to solve this covariance problem. I would appreciate some help, hint

Answer 1

Well, covariance is a measure of linear correlation... But you were asking to calculate the answer.

So, for any given $X$, the value for $Y$ will be uniformly distributed over the vertical slice through the ball. $$Y\mid X~\sim~\mathcal{U}\big[~{-}\sqrt{1-X^2},\sqrt{1-X^2}~\big]$$

And by the Law of Total Covariance:$$\mathsf{Cov}(X,Y)=\mathsf E(\mathsf{Cov}(X,Y\mid X))+\mathsf{Cov}(X,\mathsf E(Y\mid X))$$

Answer 2

Maybe a little cheeky:

$(X,Y)$ has the same distribution as $(X, -Y)$ (they are both uniform on the unit disk).

Thus, $\text{Cov}(X,Y)=\text{Cov}(X,-Y)$. But by linearity of covariance we also know $\text{Cov}(X,-Y) = -\text{Cov}(X,Y)$. Therefore,...