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  "body": "\n## Proof\n\nWe expand the covariance and use the product rule for independent variables.\n\n1. Let $X$ and $Y$ be [[independent|0E9502NT92]] real random variables with $E[X^2] \u003c \\infty$ and $E[Y^2] \u003c \\infty$.\n2. For each real $x$, $\\abs{x} \\le (1 + x^2)/2$, because $(\\abs{x} - 1)^2 \\ge 0$. The integral of nonnegative functions is monotone.\n3. So $E[\\abs{X}] \\le (1 + E[X^2])/2 \u003c \\infty$. Thus $X$ is integrable, and in the same way $Y$ is integrable. Let $m = E[X]$ and $n = E[Y]$ be their [[expected values|0DCT0QZVQ1]].\n4. By the [[product rule for independent variables|05AW1G0WE1]], $XY$ is integrable and $E[XY] = mn$.\n5. Pointwise, $(X - m)(Y - n) = XY - nX - mY + mn$.\n6. Each term on the right is integrable. By [[linearity of expectation|0P7PTC8DDE]],\n$$E[(X - m)(Y - n)] = E[XY] - n E[X] - m E[Y] + mn = E[XY] - mn.$$\n7. By the definition of the [[covariance|0DJ8W3HP2D]] and steps 4 and 6, $\\operatorname{Cov}(X, Y) = mn - mn = 0$.\n8. So $X$ and $Y$ are uncorrelated.\n\n## Idea\n\nThe covariance equals $E[XY] - E[X] E[Y]$. Independence makes the mean of the product equal to the product of the means, so this difference is $0$.\n",
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        "locator": "Uncorrelatedness and independence",
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  "name": "Proof that independent random variables are uncorrelated",
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