{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0DCT0QZVQ1"
    },
    {
      "field": "uses",
      "target": "0WF4EJP91W"
    }
  ],
  "body": "\n## Definition\n\nLet $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\\Omega, \\mathcal{F}, P)$.\n$X$ is discrete if there is a [[countable set|0ZG61TFBES]] $S \\subseteq \\mathbb{R}$ with $P(X \\in S) = 1$.\nA distribution $\\mu$ on $\\mathbb{R}$ is discrete if $\\mu(S) = 1$ for some countable set $S$.\n\n## Remarks\n\nThen $P(X \\in B) = \\sum_{x \\in B \\cap S} P(X = x)$ for each Borel set $B$. So the [[probability mass function|0WF4EJP91W]] $x \\mapsto P(X = x)$ determines the law of $X$.\nThe cumulative distribution function of a discrete random variable is a step function when the set of values has no limit points. It jumps by $P(X = x)$ at each value $x$.\nThe Bernoulli, binomial, geometric and Poisson distributions are discrete.\n\n## Explanations\n\n### L1 human\n\nA discrete random variable takes values that you can list, such as $0, 1, 2, \\dots$.\nThe number of heads in ten coin tosses and the number of calls in an hour are discrete random variables.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0X2J0VY6DN",
    "kind": "definition",
    "msc": [
      "60E05"
    ],
    "name": "Discrete random variable",
    "sources": [
      {
        "locator": "Discrete probability distribution",
        "url": "https://en.wikipedia.org/wiki/Probability_distribution"
      }
    ],
    "tag": "0J6Z6H5VQJ",
    "uses": [
      "0DKRR22AEQ",
      "0ZG61TFBES"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0J6Z6H5VQJ/",
  "links": [
    {
      "field": "defines",
      "target": "0X2J0VY6DN"
    },
    {
      "field": "uses",
      "target": "0DKRR22AEQ"
    },
    {
      "field": "uses",
      "target": "0ZG61TFBES"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0J6Z6H5VQJ.md",
  "msc": [
    "60E05"
  ],
  "name": "Discrete random variable",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0J6/0J6Z6H5VQJ.md",
  "status": null,
  "tag": "0J6Z6H5VQJ",
  "uri": "m2://0J6Z6H5VQJ",
  "url": "https://math2.tamnd.com/wiki/discrete-random-variable-definition/",
  "writing": {
    "tag": "0J6Z6H5VQJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}