{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "YNBTV2S7SR"
    }
  ],
  "body": "\n## Definition\n\nLet $\\mathcal{P} = \\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] on a measurable space $(S, \\mathcal{S})$, and let $X$ be a random element of $S$.\nLet $T = g(X)$ be a [[statistic|036756NFD6]] with values in $\\R^k$.\nThe *sampling distribution* of $T$ under $\\theta$ is the [[probability distribution|0QB913WM5E]] of $T$ when $X$ has law $P_\\theta$:\n$$P_\\theta(T \\in B) = P_\\theta\\left(g^{-1}(B)\\right)$$\nfor every Borel set $B \\subseteq \\R^k$.\n\n## Remarks\n\nExample: let $X_1, \\dots, X_n$ be i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$.\nThe sample mean $\\bar{X}_n$ is a linear combination of independent normal variables, so its sampling distribution is $\\mathcal{N}(\\mu, \\sigma^2/n)$.\nNon-example: the histogram of one observed data set is not the sampling distribution of anything; it is a single realization, while a sampling distribution ranges over all possible data.\nThe standard deviation of the sampling distribution of an estimator is its standard error.\nIn general the sampling distribution depends on $\\theta$; a statistic whose sampling distribution is the same for all $\\theta$ is [[ancillary|VBW2A4H6TD]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "656Y8FA3RY",
    "kind": "definition",
    "msc": [
      "62E15",
      "62A01"
    ],
    "name": "Sampling distribution",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sampling_distribution"
      }
    ],
    "tag": "FC9NRV5GSP",
    "uses": [
      "JZ6R7J36K5",
      "036756NFD6",
      "0QB913WM5E"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FC9NRV5GSP/",
  "links": [
    {
      "field": "defines",
      "target": "656Y8FA3RY"
    },
    {
      "field": "uses",
      "target": "JZ6R7J36K5"
    },
    {
      "field": "uses",
      "target": "036756NFD6"
    },
    {
      "field": "uses",
      "target": "0QB913WM5E"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FC9NRV5GSP.md",
  "msc": [
    "62E15",
    "62A01"
  ],
  "name": "Sampling distribution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FC9/FC9NRV5GSP.md",
  "status": null,
  "tag": "FC9NRV5GSP",
  "uri": "m2://FC9NRV5GSP",
  "url": "https://tamnd.github.io/math2/wiki/sampling-distribution-definition/",
  "writing": {
    "tag": "FC9NRV5GSP",
    "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
      }
    ]
  }
}