{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space and let $(S, \\mathcal{S})$ be a measurable space.\nLet $X_1, \\dots, X_n$ be [[random variables|0PNRKMRC55]] on $\\Omega$ with values in $S$.\nThe finite sequence $(X_1, \\dots, X_n)$ is *exchangeable* if for every [[permutation|05RWFKVTJ7]] $\\sigma$ of $\\set{1, \\dots, n}$ the random vector $(X_{\\sigma(1)}, \\dots, X_{\\sigma(n)})$ has the same [[joint distribution|0DR81R1F2B]] as $(X_1, \\dots, X_n)$.\nAn infinite sequence $X_1, X_2, \\dots$ is *exchangeable* if $(X_1, \\dots, X_n)$ is exchangeable for every $n \\ge 1$.\n\n## Remarks\n\nEquivalently, an infinite sequence is exchangeable if its joint law on $S^{\\N}$ is invariant under every permutation of $\\N$ that moves only finitely many indices.\nExample: if $X_1, X_2, \\dots$ are [[i.i.d.|00MRN0FARV]], the joint law of $(X_1, \\dots, X_n)$ is the product $\\mu^{\\otimes n}$ of the common law $\\mu$, and a product of equal factors does not change when the factors are permuted.\nAnother example: in the [[Pólya urn model|DHPXDFEEMF]], the sequence of colours drawn is exchangeable but not independent.\nNon-example: let $Y$ be a [[Bernoulli|0386Y0Z7Y7]] variable with parameter $\\frac{1}{2}$, and let $X_1 = Y$ and $X_2 = 2Y$.\nThen $P((X_1, X_2) = (1, 2)) = \\frac{1}{2}$ but $P((X_2, X_1) = (1, 2)) = 0$, so the transposition of $1$ and $2$ changes the law.\nEach $X_i$ of an exchangeable sequence has the same law, and each pair $(X_i, X_j)$ with $i \\ne j$ has the same joint law.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "4MR4YJS1ZD",
    "kind": "definition",
    "msc": [
      "60G09"
    ],
    "name": "Exchangeable random variables",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Exchangeable_random_variables"
      }
    ],
    "tag": "XKBABTE7Y2",
    "uses": [
      "0PNRKMRC55",
      "0DR81R1F2B",
      "05RWFKVTJ7",
      "00MRN0FARV"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/XKBABTE7Y2/",
  "links": [
    {
      "field": "defines",
      "target": "4MR4YJS1ZD"
    },
    {
      "field": "uses",
      "target": "0PNRKMRC55"
    },
    {
      "field": "uses",
      "target": "0DR81R1F2B"
    },
    {
      "field": "uses",
      "target": "05RWFKVTJ7"
    },
    {
      "field": "uses",
      "target": "00MRN0FARV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/XKBABTE7Y2.md",
  "msc": [
    "60G09"
  ],
  "name": "Exchangeable random variables",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XKB/XKBABTE7Y2.md",
  "status": null,
  "tag": "XKBABTE7Y2",
  "uri": "m2://XKBABTE7Y2",
  "url": "https://math2.tamnd.com/wiki/exchangeable-random-variables-definition/",
  "writing": {
    "tag": "XKBABTE7Y2",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}