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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?

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Correlation is a measure of linear dependence, so it kind of gives you an indication of how the two variables are related linearly. It doesn't capture however more complicated behaviour.

Therefore if you have $X$ and $X^2$ with $X \sim N(0,1)$, then

$$\operatorname{Cov}(X, X^2) = E(X^3) - E(X)E(X^2) = 0$$

but the two random variables are clearly dependent.

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    $\begingroup$ It's one of the reasons why when you're working with statistical data, you should always study data plots. Claims that there's no statistical relationship when there's an obvious "smile" on a scatter plot can be very misleading. $\endgroup$ – johnny Dec 2 '12 at 20:32
  • $\begingroup$ 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 $X$ $\endgroup$ – kurtkim Sep 11 '15 at 1:21
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No, it doesn't imply that. Correlation is just a one-dimensional measure, whereas dependence can take many forms. For instance, the indicator variable of the event that a normally distributed random variable is within one standard deviation of the mean is uncorrelated with the random variable itself, but is clearly not independent of it.

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