{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\lambda \u003e 0$ and let $X$ have the [[Poisson distribution|0VEFSSKX18]] with parameter $\\lambda$. Then\n$$E[X] = \\sum_{k=0}^{\\infty} k \\, e^{-\\lambda} \\frac{\\lambda^k}{k!} = \\lambda.$$\n\n## Remarks\n\nProof: the term with $k = 0$ is zero. For $k \\ge 1$, $k \\lambda^k / k! = \\lambda \\cdot \\lambda^{k-1}/(k-1)!$. So the sum is $\\lambda e^{-\\lambda} \\sum_{j \\ge 0} \\lambda^j / j! = \\lambda e^{-\\lambda} e^{\\lambda} = \\lambda$.\nThe variance of $X$ is also $\\lambda$.\nSo the parameter $\\lambda$ is the mean number of events.\n\n## Explanations\n\n### L1 human\n\nThe parameter of a Poisson distribution is its average.\nIf a shop gets on average $4$ customers per hour, a Poisson model of the number of customers in one hour uses $\\lambda = 4$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "60E05"
    ],
    "name": "Mean of the Poisson distribution",
    "role": "proposition",
    "sources": [
      {
        "locator": "Descriptive statistics",
        "url": "https://en.wikipedia.org/wiki/Poisson_distribution"
      }
    ],
    "tag": "086HEVXR9Z",
    "uses": [
      "0VEFSSKX18",
      "0DCT0QZVQ1"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/086HEVXR9Z/",
  "links": [
    {
      "field": "uses",
      "target": "0VEFSSKX18"
    },
    {
      "field": "uses",
      "target": "0DCT0QZVQ1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/086HEVXR9Z.md",
  "msc": [
    "60E05"
  ],
  "name": "Mean of the Poisson distribution",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/086/086HEVXR9Z.md",
  "status": null,
  "tag": "086HEVXR9Z",
  "uri": "m2://086HEVXR9Z",
  "url": "https://math2.tamnd.com/wiki/mean-of-the-poisson-distribution/",
  "writing": {
    "tag": "086HEVXR9Z",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}