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  "body": "\n## Definition\n\nLet $X$ be an integrable real random variable on a probability space $(\\Omega, \\mathcal{F}, P)$, and let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]].\nA conditional expectation of $X$ given $\\mathcal{G}$ is an integrable random variable $Z$ with these two properties:\n1. $Z$ is $\\mathcal{G}$-measurable.\n2. $E[Z \\, 1_G] = E[X \\, 1_G]$ for each $G \\in \\mathcal{G}$.\nSuch a $Z$ exists, and two such random variables are equal almost surely. One writes $Z = E[X \\mid \\mathcal{G}]$.\nFor a random variable $Y$, $E[X \\mid Y] = E[X \\mid \\sigma(Y)]$, where $\\sigma(Y)$ is the σ-algebra generated by $Y$.\n\n## Remarks\n\nThe existence follows from the Radon–Nikodym theorem. For $E[X^2] \u003c \\infty$, $E[X \\mid \\mathcal{G}]$ is the orthogonal projection of $X$ onto $L^2(\\mathcal{G})$.\nFor a discrete $Y$, $E[X \\mid Y] = g(Y)$ with $g(y) = E[X \\, 1_{\\{Y = y\\}}] / P(Y = y)$ when $P(Y = y) \u003e 0$. This $g(y)$ is the expectation of $X$ under the [[conditional probability|0FJB06AD2P]] $P(\\cdot \\mid Y = y)$.\nThe conditional expectation is linear and monotone. If $W$ is $\\mathcal{G}$-measurable and $WX$ is integrable, then $E[WX \\mid \\mathcal{G}] = W \\, E[X \\mid \\mathcal{G}]$.\nIf $X$ is independent of $\\mathcal{G}$, then $E[X \\mid \\mathcal{G}] = E[X]$.\n\n## Explanations\n\n### L1 human\n\n$E[X \\mid Y]$ is the average value of $X$ when you know $Y$. It changes with $Y$, so it is a random variable.\nFor two dice with results $D_1$ and $D_2$, $E[D_1 + D_2 \\mid D_1] = D_1 + 3.5$.\n",
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