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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[independent|0E9502NT92]] integrable real random variables.\nThen $XY$ is integrable and\n$$E[XY] = E[X] \\, E[Y].$$\n\n## Remarks\n\nThe proof uses that the joint law of $(X, Y)$ is the [[product of the marginal laws|08QCA2F1NX]], and then Fubini's theorem.\nFor nonnegative independent $X$ and $Y$, the equality holds in $[0, \\infty]$ without integrability.\nThe converse is false: $E[XY] = E[X] \\, E[Y]$ means only that $X$ and $Y$ are uncorrelated.\nMore generally, $E[g(X) \\, h(Y)] = E[g(X)] \\, E[h(Y)]$ for measurable $g$ and $h$ with $g(X)$ and $h(Y)$ integrable.\n\n## Explanations\n\n### L1 human\n\nFor independent variables, the average of a product is the product of the averages.\nFor two fair dice, the expected product of the two results is $3.5 \\cdot 3.5 = 12.25$.\n",
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