{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "09J1H07VAH"
    },
    {
      "field": "uses",
      "target": "0NNRNGKMFM"
    },
    {
      "field": "uses",
      "target": "0STK8KEY9G"
    },
    {
      "field": "uses",
      "target": "0ZT24EC90K"
    }
  ],
  "body": "\n## Definition\n\nLet $X$ be a real random variable and let $k \\ge 1$ be an integer with $E[|X|^k] \u003c \\infty$.\nThe $k$-th moment of $X$ is\n$$m_k = E[X^k].$$\nThe $k$-th absolute moment is $E[|X|^k]$.\n\n## Remarks\n\nBy the [[law of the unconscious statistician|01ZWX875AT]], $m_k = \\int x^k \\, \\mu_X(dx)$. So the moments depend only on the law $\\mu_X$ of $X$.\nThe first moment is the mean. The variance is $m_2 - m_1^2$.\nIf $E[|X|^k] \u003c \\infty$, then $E[|X|^j] \u003c \\infty$ for each $j \\le k$.\nThe [[Cauchy distribution|0NA6Q82YR7]] has no moments of order $k \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nMoments are numbers that sum up the shape of a distribution.\nThe first gives the center, the second with the first gives the spread, and the higher ones describe asymmetry and the weight of the tails.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "01XAHPADBA",
    "kind": "definition",
    "msc": [
      "60E05"
    ],
    "name": "Moment",
    "sources": [
      {
        "locator": "Formulation",
        "url": "https://en.wikipedia.org/wiki/Moment_(mathematics)"
      }
    ],
    "tag": "0PH0VHTQS0",
    "uses": [
      "0DCT0QZVQ1",
      "01ZWX875AT"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0PH0VHTQS0/",
  "links": [
    {
      "field": "defines",
      "target": "01XAHPADBA"
    },
    {
      "field": "uses",
      "target": "0DCT0QZVQ1"
    },
    {
      "field": "uses",
      "target": "01ZWX875AT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0PH0VHTQS0.md",
  "msc": [
    "60E05"
  ],
  "name": "Moment",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0PH/0PH0VHTQS0.md",
  "status": null,
  "tag": "0PH0VHTQS0",
  "uri": "m2://0PH0VHTQS0",
  "url": "https://math2.tamnd.com/wiki/moment-definition/",
  "writing": {
    "tag": "0PH0VHTQS0",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}