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  "body": "\n## Statement\n\nLet $X$ be an integrable real random variable and let $b \\in \\mathbb{R}$. Then\n$$\\operatorname{Var}(X + b) = \\operatorname{Var}(X).$$\n\n## Remarks\n\nProof: $E[X + b] = E[X] + b$ by [[linearity of expectation|0P7PTC8DDE]], so $(X + b) - E[X + b] = X - E[X]$.\nTogether with the [[scaling rule|0KVQ9D86BM]], this gives $\\operatorname{Var}(aX + b) = a^2 \\operatorname{Var}(X)$ for all real $a$ and $b$.\n\n## Explanations\n\n### L1 human\n\nIf you add the same constant to every value, the whole distribution moves, but its spread does not change.\n",
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