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?