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  "body": "\n## Proof\n\nWe show that the deviation from the mean does not change.\n\n1. Let $X$ be an integrable real random variable, let $b \\in \\R$, and let $m = E[X]$.\n2. The constant $b$ is integrable with $E[b] = b$. By [[linearity of expectation|0P7PTC8DDE]], $X + b$ is integrable and $E[X + b] = m + b$.\n3. Pointwise, $(X + b) - (m + b) = X - m$.\n4. So $((X + b) - E[X + b])^2 = (X - m)^2$ as random variables.\n5. By the definition of the [[variance|007QED3E0Y]], $\\operatorname{Var}(X + b) = E[(X - m)^2] = \\operatorname{Var}(X)$, as an equality in $[0, \\infty]$.\n\n## Idea\n\nAdding $b$ moves every value and the mean by the same amount. The distances to the mean stay the same, so the variance stays the same.\n",
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