{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PMDXYNPE7D"
    }
  ],
  "body": "\n## Summary\n\nA real random variable is sub-Gaussian when its tails decay at least as fast as those of a [[normal distribution|05AQE0QR0D]], that is, like $e^{-c t^2}$.\nNormal variables and bounded variables are sub-Gaussian.\nSums of independent sub-Gaussian variables are again sub-Gaussian, with explicit constants, and this gives concentration bounds of [[Hoeffding|0WPDNXGSSN]] type.\nThe notion is central in high-dimensional probability, random matrix theory and statistical learning.\n\n## Explanations\n\n### L1 human\n\nA sub-Gaussian variable is one that is very unlikely to be far from $0$: the chance of exceeding $t$ drops like $e^{-c t^2}$, as for a bell curve.\nA fair $\\pm 1$ coin is sub-Gaussian, since it never exceeds $1$ at all.\nAn [[exponential|0E99F51GVB]] variable is not, since $P(X \u003e t) = e^{-t}$ decays only like $e^{-t}$.\nThe average of many independent sub-Gaussian variables is sharply concentrated around its mean.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "60E15",
      "60E05"
    ],
    "name": "Sub-Gaussian distribution",
    "same_as": {
      "wikidata": "Q25303973",
      "wikipedia": "Sub-Gaussian distribution"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sub-Gaussian_distribution"
      }
    ],
    "specializes": [
      "0DVDCMA2DK"
    ],
    "tag": "X9R1PCKJ0R"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/X9R1PCKJ0R/",
  "links": [
    {
      "field": "specializes",
      "target": "0DVDCMA2DK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/X9R1PCKJ0R.md",
  "msc": [
    "60E15",
    "60E05"
  ],
  "name": "Sub-Gaussian distribution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X9R/X9R1PCKJ0R.md",
  "status": null,
  "tag": "X9R1PCKJ0R",
  "uri": "m2://X9R1PCKJ0R",
  "url": "https://math2.tamnd.com/wiki/sub-gaussian-distribution/",
  "writing": {
    "tag": "X9R1PCKJ0R",
    "kind": "concept",
    "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
      }
    ]
  }
}