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2011-05-12T13:33:23Z

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In [[probability theory]], '''Gauss's inequality''' (or the '''Gauss inequality''') gives an upper bound on the probability that a [[unimodal]] [[random variable]] lies more than any given distance from its [[mode (statistics)|mode]].

Let ''X'' be a unimodal random variable with mode ''m'', and let ''τ'' 2 be the [[expected value]] of (''X'' − ''m'')2. (''τ'' 2 can also be expressed as (''μ'' − ''m'')2 + ''σ'' 2, where ''μ'' and ''σ'' are the mean and [[standard deviation]] of ''X''.) Then for any positive value of ''k'',

:<math>
\Pr(\mid X - m \mid > k) \leq \begin{cases}
\left( \frac{2\tau}{3k} \right)^2 & \text{if } k \geq \frac{2\tau}{\sqrt{3}} \\[6pt]
1 - \frac{k}{\tau\sqrt{3}} & \text{if } 0 \leq k \leq \frac{2\tau}{\sqrt{3}}.
\end{cases}</math>

The theorem was first proved by [[Carl Friedrich Gauss]] in 1823.

==See also==
*[[Vysochanskiï–Petunin inequality]], a similar result for the distance from the mean rather than the mode
*[[Chebyshev's inequality]], concerns distance from the mean without requiring unimodality

==References==
*{{cite journal|last=Gauss|first=C. F.|authorlink=Carl Friedrich Gauss|date=1823|title=Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior|journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores|volume=5}}
*{{cite book|title=A Dictionary of Statistics| publisher=Oxford University Press| last= Upton | first = Graham | coauthors = Cook, Ian| year=2008|chapter=Gauss inequality| url=http://www.answers.com/topic/gauss-inequality}}
*{{Cite journal
| doi = 10.2307/2684690
| title = Chebyshev inequalities for unimodal distributions
| year = 1997
| journal = [[American Statistician]]
| volume = 51
| issue = 1
| pages = 34–40
| last1 = Sellke | first1 = T.M.
| last2 = Sellke | first2 = S.H.
| publisher = American Statistical Association
| jstor = 2684690
}}
*{{Cite journal
| doi = 10.2307/2684253
| title = The Three Sigma Rule
| year = 1994
| author = Pukelsheim, F.
| journal = American Statistician
| volume = 48
| issue = 2
| pages = 88–91
| publisher = American Statistical Association
| jstor = 2684253
}}

[[Category:Probabilistic inequalities]]

Precursor (physics) - Revision history

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