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  "body": "\n## Definition\n\nLet $X$ be a real [[random variable|0DKRR22AEQ]] on a [[probability space|0ZYBZ34RT2]] $(\\Omega, \\mathcal{F}, P)$.\nIf $X \\ge 0$, the expected value of $X$ is the Lebesgue integral\n$$E[X] = \\int_\\Omega X \\, dP \\in [0, \\infty].$$\nIn general, write $X = X^+ - X^-$ with $X^+ = \\max(X, 0)$ and $X^- = \\max(-X, 0)$. If $E[|X|] \u003c \\infty$, then $X$ is integrable and $E[X] = E[X^+] - E[X^-]$.\n\n## Remarks\n\nFor a [[discrete random variable|0J6Z6H5VQJ]] with values $x_1, x_2, \\dots$, $E[X] = \\sum_i x_i \\, P(X = x_i)$ when the series converges absolutely.\nFor a [[random variable with density|0C21RZG5V2]] $f$, $E[X] = \\int_{-\\infty}^{\\infty} x f(x) \\, dx$ when $\\int |x| f(x) \\, dx \u003c \\infty$.\nOther notations are $\\mathbb{E}[X]$, $\\mathbb{E}X$ and $\\mu_X$. The expected value depends only on the law of $X$.\nIf $E[X^+] = \\infty$ and $E[X^-] \u003c \\infty$, one sets $E[X] = +\\infty$. If both are infinite, $E[X]$ is not defined.\n\n## Explanations\n\n### L1 human\n\nThe expected value is the long-run average of many independent repetitions of the experiment.\nFor a fair die, it is $(1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5$. This value never appears on the die, but it is the center of mass of the distribution.\n",
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