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Cantelli's inequality — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Cantelli%27s_inequality
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T18:28:51.773Z
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Cantelli's inequality — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Cantelli%27s_inequality (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_fddca8ccf2eb (Metraipolis archive dc_fddca8ccf2eb, sha256 4d5f96f67b5176b37e98f151ac5eda006a6c216b9877fc446078a65eaeede864)

BibTeX

@misc{dcfddca8ccf2eb,
  title = {Cantelli's inequality — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Cantelli%27s_inequality},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_fddca8ccf2eb; sha256 4d5f96f67b5176b37e98f151ac5eda006a6c216b9877fc446078a65eaeede864},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Cantelli%27s_inequality. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for λ > 0 , {\displaystyle \lambda >0,} Pr ( X − E [ X ] ≥ λ ) ≤ σ 2 σ 2 + λ 2 , {\displaystyle \Pr(X-\mathbb {E} [X]\geq \lambda )\leq {\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}},} where X {\displaystyle X} is a real-valued random variable, Pr {\displaystyle \Pr } is the probability measure, E [ X ] {\displaystyle \mathbb {E} [X]} is the expected value of X {\displaystyle X} , σ 2 {\displaystyle \sigma ^{2}} is the variance of X {\displaystyle X} . Applying the Cantelli inequality to − X {\displaystyle -X} gives a bound on the lower tail, Pr ( X − E [ X ] ≤ − λ ) ≤ σ 2 σ 2 + λ 2 . {\displaystyle \Pr(X-\mathbb {E} [X]\leq -\lambda )\leq {\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}}.} While the inequality is often attributed to Francesco Paolo Cantelli who published it in 1928, it originates in Chebyshev's work of 1874. When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. The Chebyshev inequality has "higher moments versions" and "vector versions", and so does the Cantelli inequality.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «probability theory»).
Fonte: Wikipedia in inglese, voce «Cantelli's inequality» — https://en.wikipedia.org/wiki/Cantelli%27s_inequality
Wikidata Q3711841: «generalization of Chebyshev's inequality for a single tail» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q3711841
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Cantelli%27s_inequality?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1301274414, consultata il 2026-10-07T17:57:24.055Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 807e0ce585e24cc5ea340657b57ededf61d07df35be1f1eb1289a2ded91a4566