TITLE: Does zero covariance imply independence of random variables here?
QUESTION [7 upvotes]: I have two random variables $X$ and $Y$. Both are distributed according to $N(0,1)$. If their covariance is 0, are they independent?
I know that this is not true for other distributions, say the Wikipedia example: $X$ chosen uniformly in $[-1,1]$ and $Y=X^2$.

REPLY [7 votes]: http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent