# Why can 2 uncorrelated random variables be dependent?

I recently learned that two independent random variables $X$ and $Y$ must have a covariance of 0. That means that the correlation between them is also 0.

However, apparently, the converse is not true. 2 random variables $X$ and $Y$ can have a correlation of 0, yet still be dependent. I don't understand why this is. Doesn't a correlation of 0 imply that the random variables do not affect each other?

Therefore if you have $X$ and $X^2$ with $X \sim N(0,1)$, then
$$\operatorname{Cov}(X, X^2) = E(X^3) - E(X)E(X^2) = 0$$
• I am not sure whether you can see this or not, however, how can we guarantee $E(X^3) = 0$ i.e., 3rd moment of $X$ – kurtkim Sep 11 '15 at 1:21