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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[independent|0E9502NT92]] real random variables with $E[X^2] \u003c \\infty$ and $E[Y^2] \u003c \\infty$.\nThen $\\operatorname{Cov}(X, Y) = 0$, so $X$ and $Y$ are uncorrelated.\n\n## Remarks\n\nProof: by the [[product rule for independent variables|05AW1G0WE1]], $E[XY] = E[X] \\, E[Y]$. So $\\operatorname{Cov}(X, Y) = E[XY] - E[X] \\, E[Y] = 0$.\nThe converse is false. The node [[uncorrelated random variables that are not independent|0C6J8BQXVR]] gives a counterexample.\nFor a pair $(X, Y)$ with a joint multivariate normal law, uncorrelated does imply independent.\n\n## Explanations\n\n### L1 human\n\nIf two variables do not influence each other at all, they have no linear relation either.\nBut \"no linear relation\" is a much weaker property than independence.\n",
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