{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "08QCA2F1NX"
    },
    {
      "field": "uses",
      "target": "0AXP2Z5N9H"
    },
    {
      "field": "uses",
      "target": "0YVDC5PYDW"
    }
  ],
  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be real [[random variables|0DKRR22AEQ]] on one probability space $(\\Omega, \\mathcal{F}, P)$.\nTheir joint distribution is the [[law|0TST4MHA3A]] of the random vector $(X_1, \\dots, X_n) \\colon \\Omega \\to \\mathbb{R}^n$. It is the probability measure on the Borel sets of $\\mathbb{R}^n$ with\n$$\\mu(B) = P((X_1, \\dots, X_n) \\in B).$$\nTheir joint cumulative distribution function is $F(x_1, \\dots, x_n) = P(X_1 \\le x_1, \\dots, X_n \\le x_n)$.\n\n## Remarks\n\nFor discrete variables, the joint probability mass function $p(x_1, \\dots, x_n) = P(X_1 = x_1, \\dots, X_n = x_n)$ determines the joint distribution.\nThe variables have a joint density $f$ if $\\mu(B) = \\int_B f(x) \\, dx$ for each Borel set $B \\subseteq \\mathbb{R}^n$.\nThe joint distribution determines the [[marginal distributions|0AXP2Z5N9H]], but the marginal distributions do not determine the joint distribution.\nFor two rolls of a fair die, the joint mass function of the two results is $1/36$ at each pair $(i, j)$ with $1 \\le i, j \\le 6$.\n\n## Explanations\n\n### L1 human\n\nThe joint distribution describes several random quantities together, not one by one.\nFor example, it says how likely it is that a person is tall and also heavy.\nIt gives more than the chance of each of the two alone.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0DR81R1F2B",
    "kind": "definition",
    "msc": [
      "62H05"
    ],
    "name": "Joint probability distribution",
    "sources": [
      {
        "locator": "Joint cumulative distribution function",
        "url": "https://en.wikipedia.org/wiki/Joint_probability_distribution"
      }
    ],
    "tag": "0QAR0XDZH4",
    "uses": [
      "0DKRR22AEQ",
      "0TST4MHA3A",
      "0C81YBVTWH"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0QAR0XDZH4/",
  "links": [
    {
      "field": "defines",
      "target": "0DR81R1F2B"
    },
    {
      "field": "uses",
      "target": "0DKRR22AEQ"
    },
    {
      "field": "uses",
      "target": "0TST4MHA3A"
    },
    {
      "field": "uses",
      "target": "0C81YBVTWH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0QAR0XDZH4.md",
  "msc": [
    "62H05"
  ],
  "name": "Joint probability distribution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0QA/0QAR0XDZH4.md",
  "status": null,
  "tag": "0QAR0XDZH4",
  "uri": "m2://0QAR0XDZH4",
  "url": "https://math2.tamnd.com/wiki/joint-probability-distribution-definition/",
  "writing": {
    "tag": "0QAR0XDZH4",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}