Independent random variables are uncorrelated

Statement

Let $X$ and $Y$ be independent real random variables with $E[X^2] < \infty$ and $E[Y^2] < \infty$. Then $\operatorname{Cov}(X, Y) = 0$, so $X$ and $Y$ are uncorrelated.

Remarks

Proof: by the product rule for independent variables, $E[XY] = E[X] \, E[Y]$. So $\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y] = 0$. The converse is false. The node uncorrelated random variables that are not independent gives a counterexample. For a pair $(X, Y)$ with a joint multivariate normal law, uncorrelated does imply independent.

Explanations

L1 human

If two variables do not influence each other at all, they have no linear relation either. But "no linear relation" is a much weaker property than independence.