{
  "aliases": [
    "law-of-rare-events",
    "poisson-approximation-to-the-binomial"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\lambda \u003e 0$, and let $p_n \\in [0, 1]$ satisfy $n p_n \\to \\lambda$ as $n \\to \\infty$. Let $S_n \\sim \\operatorname{Bin}(n, p_n)$.\nThen for each integer $k \\ge 0$,\n$$\\lim_{n \\to \\infty} P(S_n = k) = \\lim_{n \\to \\infty} \\binom{n}{k} p_n^k (1 - p_n)^{n - k} = e^{-\\lambda} \\frac{\\lambda^k}{k!}.$$\nSo $S_n$ converges [[in distribution|0JDAQHWA9N]] to the [[Poisson distribution|0VEFSSKX18]] with parameter $\\lambda$.\n\n## Remarks\n\nProof: write $\\binom{n}{k} p_n^k = \\frac{n(n-1)\\cdots(n-k+1)}{n^k} \\cdot \\frac{(n p_n)^k}{k!}$. The first factor tends to $1$, and $(1 - p_n)^{n-k} \\to e^{-\\lambda}$.\nIn practice, $\\operatorname{Bin}(n, p)$ is close to $\\operatorname{Poisson}(np)$ when $n$ is large and $p$ is small.\nLe Cam's inequality gives an explicit bound on the error of this approximation.\n\n## Explanations\n\n### L1 human\n\nThe number of rare events among many independent chances is approximately Poisson.\nExamples are the number of misprints on a page, or the number of large insurance claims in a year.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "law-of-rare-events",
      "poisson-approximation-to-the-binomial"
    ],
    "kind": "statement",
    "msc": [
      "60F05"
    ],
    "name": "Poisson limit theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7208500",
      "wikipedia": "Poisson limit theorem"
    },
    "sources": [
      {
        "locator": "Theorem",
        "url": "https://en.wikipedia.org/wiki/Poisson_limit_theorem"
      }
    ],
    "tag": "0XKTZRWTXX",
    "uses": [
      "0VRWZDWW6B",
      "0VEFSSKX18",
      "0JDAQHWA9N"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0XKTZRWTXX/",
  "links": [
    {
      "field": "uses",
      "target": "0VRWZDWW6B"
    },
    {
      "field": "uses",
      "target": "0VEFSSKX18"
    },
    {
      "field": "uses",
      "target": "0JDAQHWA9N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0XKTZRWTXX.md",
  "msc": [
    "60F05"
  ],
  "name": "Poisson limit theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0XK/0XKTZRWTXX.md",
  "status": null,
  "tag": "0XKTZRWTXX",
  "uri": "m2://0XKTZRWTXX",
  "url": "https://math2.tamnd.com/wiki/poisson-limit-theorem/",
  "writing": {
    "tag": "0XKTZRWTXX",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}