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  "body": "\n## Proof\n\nWe expand the square of the centred sum and use linearity of expectation.\n\n1. Let $X_1, \\dots, X_n$ be real random variables on one probability space with $E[X_i^2] \u003c \\infty$ for each $i$. Let $S = X_1 + \\dots + X_n$.\n2. For each real $x$, $\\abs{x} \\le (1 + x^2)/2$, because $(\\abs{x} - 1)^2 \\ge 0$.\n3. The integral of nonnegative functions is monotone. So by step 2, $E[\\abs{X_i}] \\le (1 + E[X_i^2])/2 \u003c \\infty$, and each $X_i$ is integrable, with [[expected value|0DCT0QZVQ1]] $m_i = E[X_i]$.\n4. Let $Y_i = X_i - m_i$. Since $(x - m)^2 \\le 2x^2 + 2m^2$ for real $x$ and $m$, we get $E[Y_i^2] \\le 2E[X_i^2] + 2m_i^2 \u003c \\infty$.\n5. For real $u$ and $v$, $\\abs{uv} \\le (u^2 + v^2)/2$. So $\\abs{Y_i Y_j} \\le (Y_i^2 + Y_j^2)/2$, and each product $Y_i Y_j$ is integrable.\n6. By [[linearity of expectation|0P7PTC8DDE]], $S$ is integrable and $E[S] = m_1 + \\dots + m_n$. So $S - E[S] = Y_1 + \\dots + Y_n$.\n7. Expanding the square of a finite sum gives, pointwise,\n$$(S - E[S])^2 = \\sum_{i=1}^n Y_i^2 + 2 \\sum_{1 \\le i \u003c j \\le n} Y_i Y_j.$$\n8. All terms on the right are integrable by steps 4 and 5. By linearity of expectation,\n$$E[(S - E[S])^2] = \\sum_{i=1}^n E[Y_i^2] + 2 \\sum_{1 \\le i \u003c j \\le n} E[Y_i Y_j].$$\n9. By the definition of the [[variance|007QED3E0Y]], the left side is $\\operatorname{Var}(S)$ and $E[Y_i^2] = \\operatorname{Var}(X_i)$.\n10. By the definition of the [[covariance|0DJ8W3HP2D]], $E[Y_i Y_j] = E[(X_i - m_i)(X_j - m_j)] = \\operatorname{Cov}(X_i, X_j)$.\n11. Steps 8 to 10 give\n$$\\operatorname{Var}\\left(\\sum_{i=1}^n X_i\\right) = \\sum_{i=1}^n \\operatorname{Var}(X_i) + 2 \\sum_{1 \\le i \u003c j \\le n} \\operatorname{Cov}(X_i, X_j).$$\n12. For $n = 2$ with $X_1 = X$ and $X_2 = Y$, the only pair is $(1, 2)$. So $\\operatorname{Var}(X + Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y) + 2 \\operatorname{Cov}(X, Y)$.\n\n## Idea\n\nThe deviation of the sum from its mean is the sum of the deviations of the terms. Squaring a sum gives the squares plus twice the cross products. Their expectations are the variances and the covariances.\n",
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