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  "body": "\n## Statement\n\nLet $X$ be a real random variable with finite [[variance|007QED3E0Y]] and let $a \\in \\mathbb{R}$. Then\n$$\\operatorname{Var}(aX) = a^2 \\operatorname{Var}(X).$$\n\n## Remarks\n\nProof: $E[aX] = a E[X]$ by [[linearity of expectation|0P7PTC8DDE]]. So $(aX - E[aX])^2 = a^2 (X - E[X])^2$, and one takes expectations.\nFor the standard deviation, the formula becomes $\\sigma_{aX} = |a| \\, \\sigma_X$.\nIn particular, $\\operatorname{Var}(-X) = \\operatorname{Var}(X)$.\n\n## Explanations\n\n### L1 human\n\nIf you measure in centimeters instead of meters, every value is multiplied by $100$. The variance is then multiplied by $100^2 = 10000$, because it is measured in squared units.\n",
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