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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be real random variables on one probability space with $E[X_i^2] \u003c \\infty$ for each $i$. Then\n$$\\operatorname{Var}\\left(\\sum_{i=1}^n X_i\\right) = \\sum_{i=1}^n \\operatorname{Var}(X_i) + 2 \\sum_{1 \\le i \u003c j \\le n} \\operatorname{Cov}(X_i, X_j).$$\nIn particular, $\\operatorname{Var}(X + Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y) + 2 \\operatorname{Cov}(X, Y)$.\n\n## Remarks\n\nProof: $\\operatorname{Var}(S) = \\operatorname{Cov}(S, S)$ for $S = \\sum_i X_i$. Expand by the bilinearity of the [[covariance|0DJ8W3HP2D]]: $\\operatorname{Cov}(S, S) = \\sum_{i, j} \\operatorname{Cov}(X_i, X_j)$.\nAlso $\\operatorname{Var}(X - Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y) - 2 \\operatorname{Cov}(X, Y)$.\nMore generally, $\\operatorname{Var}\\left(\\sum_i a_i X_i\\right) = \\sum_{i, j} a_i a_j \\operatorname{Cov}(X_i, X_j)$ for real $a_i$.\n\n## Explanations\n\n### L1 human\n\nThe spread of a sum is the sum of the spreads plus a correction for how the terms move together.\nIf the terms tend to move in the same direction, the sum spreads more. If they tend to compensate each other, it spreads less.\n",
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