{
  "aliases": [
    "tower-property",
    "law-of-iterated-expectations",
    "adam-law"
  ],
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  "body": "\n## Statement\n\nLet $X$ be an integrable real random variable on $(\\Omega, \\mathcal{F}, P)$ and let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a σ-algebra. Then\n$$E\\left[E[X \\mid \\mathcal{G}]\\right] = E[X].$$\nIn particular, $E[E[X \\mid Y]] = E[X]$ for each random variable $Y$ on the same space.\n\n## Remarks\n\nProof: put $G = \\Omega$ in the defining property of the [[conditional expectation|0YAMJRK04E]].\nFor a partition of $\\Omega$ into events $A_1, A_2, \\dots$ with $P(A_i) \u003e 0$, it gives $E[X] = \\sum_i E[X \\mid A_i] \\, P(A_i)$. This extends the [[law of total probability|0E5G5FMT2R]].\nMore generally, for σ-algebras $\\mathcal{H} \\subseteq \\mathcal{G}$, $E[E[X \\mid \\mathcal{G}] \\mid \\mathcal{H}] = E[X \\mid \\mathcal{H}]$. This is the tower property.\n\n## Explanations\n\n### L1 human\n\nTo find an overall average, you can first average inside each group and then average the group averages, with the group sizes as weights.\nIf $60\\%$ of the students have mean score $70$ and $40\\%$ have mean score $80$, the overall mean is $0.6 \\cdot 70 + 0.4 \\cdot 80 = 74$.\n",
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    "name": "Law of total expectation",
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        "locator": "Proof in the general case",
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  "name": "Law of total expectation",
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