{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $d \\ge 1$ and $\\lambda \u003e 0$.\nLet $\\Phi = \\set{x_1, x_2, \\ldots}$ be a homogeneous [[Poisson point process|05TG0SE4HT]] in $\\R^d$ with intensity $\\lambda$; its points are the *germs*.\nLet $K_1, K_2, \\ldots$ be [[independent and identically distributed|00MRN0FARV]] random nonempty [[compact sets|00R0X1WGPX]] in $\\R^d$, independent of $\\Phi$; they are the *grains*.\nLet $K$ denote a random set with their common law.\nLet $V_d$ be Lebesgue measure on $\\R^d$, and let $B$ be the [[closed unit ball|05DSV40EB9]] centred at $0$.\nAssume $E[V_d(K + B)] \u003c \\infty$, where $K + B = \\set{k + b : k \\in K, b \\in B}$.\nThe *Boolean model* with these germs and grains is the random set\n$$\\Xi = \\bigcup_{i \\ge 1} (x_i + K_i).$$\n\n## Remarks\n\nUnder the integrability assumption, only finitely many grains $x_i + K_i$ meet each bounded set almost surely, so $\\Xi$ is a random closed set.\nIts law is invariant under translations of $\\R^d$.\n\nExample: let $d = 2$ and let every grain be the closed disc of radius $r \u003e 0$ centred at $0$.\nThen $V_2(K + B) = \\pi (r + 1)^2$ is finite, so the assumption holds.\nA point $y$ is uncovered exactly when $\\Phi$ has no point in the disc of radius $r$ around $y$, which has probability $e^{-\\lambda \\pi r^2}$.\nSo the covered fraction of the plane is $p = 1 - e^{-\\lambda \\pi r^2}$.\nIn general, $P(y \\in \\Xi) = 1 - e^{-\\lambda E[V_d(K)]}$ for every $y \\in \\R^d$.\n\nNon-example: put the same discs at the points of the lattice $\\Z^2$ instead of a Poisson process.\nThe union is a union of discs, but the germs do not form a Poisson point process, so it is not a Boolean model.\n\nThe Boolean model is also called the Poisson germ-grain model.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "NK3CZTVEJD",
    "kind": "definition",
    "msc": [
      "60D05",
      "60G55"
    ],
    "name": "Boolean model",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Boolean_model_(probability_theory)"
      }
    ],
    "tag": "3PZ3TMW86C",
    "uses": [
      "05TG0SE4HT",
      "00R0X1WGPX",
      "00MRN0FARV",
      "05DSV40EB9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/3PZ3TMW86C/",
  "links": [
    {
      "field": "defines",
      "target": "NK3CZTVEJD"
    },
    {
      "field": "uses",
      "target": "05TG0SE4HT"
    },
    {
      "field": "uses",
      "target": "00R0X1WGPX"
    },
    {
      "field": "uses",
      "target": "00MRN0FARV"
    },
    {
      "field": "uses",
      "target": "05DSV40EB9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3PZ3TMW86C.md",
  "msc": [
    "60D05",
    "60G55"
  ],
  "name": "Boolean model",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3PZ/3PZ3TMW86C.md",
  "status": null,
  "tag": "3PZ3TMW86C",
  "uri": "m2://3PZ3TMW86C",
  "url": "https://math2.tamnd.com/wiki/boolean-model-definition/",
  "writing": {
    "tag": "3PZ3TMW86C",
    "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
      }
    ]
  }
}