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  "body": "\n## Statement\n\nLet $X$ be a real random variable with $E[X^2] \u003c \\infty$. Then\n$$\\operatorname{Var}(X) = E[X^2] - (E[X])^2.$$\n\n## Remarks\n\nProof: with $\\mu = E[X]$, $(X - \\mu)^2 = X^2 - 2\\mu X + \\mu^2$. By [[linearity of expectation|0P7PTC8DDE]], $E[(X - \\mu)^2] = E[X^2] - 2\\mu^2 + \\mu^2 = E[X^2] - \\mu^2$.\nAs a corollary, $(E[X])^2 \\le E[X^2]$.\nFor a fair die, $E[X^2] = 91/6$ and $E[X] = 7/2$, so $\\operatorname{Var}(X) = 91/6 - 49/4 = 35/12$.\nIn floating-point arithmetic, this formula can lose much precision when $E[X^2]$ and $(E[X])^2$ are close.\n\n## Explanations\n\n### L1 human\n\nThis formula is often the fastest way to compute a variance: find the average of the squares and subtract the square of the average.\n",
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