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  "body": "\n## Statement\n\nLet $X$ and $Y$ be integrable real random variables on one probability space, and let $a, b \\in \\mathbb{R}$.\nThen $aX + bY$ is integrable and\n$$E[aX + bY] = a \\, E[X] + b \\, E[Y].$$\n\n## Remarks\n\nThe statement is the linearity of the Lebesgue integral. It holds for all integrable $X$ and $Y$, also when they are not independent.\nBy induction, $E[X_1 + \\dots + X_n] = E[X_1] + \\dots + E[X_n]$.\nA standard use is the method of indicators. If $N$ counts how many of the events $A_1, \\dots, A_n$ occur, then $N = \\sum_i 1_{A_i}$ and $E[N] = \\sum_i P(A_i)$.\nFor nonnegative random variables, additivity $E[X + Y] = E[X] + E[Y]$ holds without integrability, in $[0, \\infty]$.\n\n## Explanations\n\n### L1 human\n\nThe average of a sum is the sum of the averages, always.\nIf a random permutation of $n$ letters goes into $n$ envelopes, each letter is in its own envelope with probability $1/n$. So the expected number of correct letters is $n \\cdot 1/n = 1$.\n",
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    "name": "Linearity of expectation",
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  "name": "Linearity of expectation",
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