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  "body": "\n## Statement\n\nLet $X \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2)$ and $Y \\sim \\mathcal{N}(\\mu_2, \\sigma_2^2)$ be [[independent|0E9502NT92]], with $\\sigma_1^2, \\sigma_2^2 \u003e 0$. Then\n$$X + Y \\sim \\mathcal{N}(\\mu_1 + \\mu_2, \\, \\sigma_1^2 + \\sigma_2^2).$$\n\n## Remarks\n\nProof with [[characteristic functions|0NS0TDA4E2]]: $\\varphi_{X+Y}(t) = \\varphi_X(t) \\varphi_Y(t) = \\exp(i(\\mu_1 + \\mu_2)t - (\\sigma_1^2 + \\sigma_2^2) t^2 / 2)$. By the [[uniqueness theorem|0T8BYAFEXX]], this fixes the law.\nBy induction, a sum of $n$ independent normal variables is normal. Its mean is the sum of the means and its variance is the sum of the variances.\nWith the rule for $aX + b$, every linear combination $a_1 X_1 + \\dots + a_n X_n$ of independent normal variables is normal, if some $a_i \\ne 0$.\nThe independence condition is necessary. Two normal variables that are not jointly normal can have a sum that is not normal.\n\n## Explanations\n\n### L1 human\n\nAdding independent normal errors gives another normal error. The means add, and the variances add, but the standard deviations do not add.\n",
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