- Vysochanskiï-Petunin inequality
In
probability theory, the Vysochanskij-Petunin inequality gives a lower bound for the probability that arandom variable with finitevariance lies within a certain number ofstandard deviation s of the variable's mean, or equivalently an upper bound for the probability that it lies further away. The sole restriction the distribution is that it be unimodal and have finitevariance . (This implies that it is acontinuous probability distribution except at the mode, which may have a non-zero probability.)The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle"."Theorem." Let "X" be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ2. Then, for any λ > √(8/3) = 1.63299...
:P(left|X-mu ight|geq lambdasigma)leqfrac{4}{9lambda^2}.
Furthermore, the limit is attained (that is, the probability is equal to 4/(9 λ2)) for a random variable having a probability 1 − 4/(3 λ2) of being exactly equal to the mean, and which, when it is not equal to the mean, is distributed uniformly in an interval centred on the mean. When λ is less than √(8/3), there exist unsymmetric distributions for which the 4/(9 λ2) limit is exceeded.
The theorem refines Chebyshëv's inequality by including the factor of 4/9, made possible by the condition that the distribution be unimodal.
It is common, in the construction of
control chart s and other statistical heuristics, to set λ = 3, corresponding to an upper probability bound of 4/81 = 0.04938, and to construct "3-sigma" limits to bound "nearly all" (i.e. 95%) of the values of aprocess output. Without unimodality Chebyshev's inequality would give a looser bound of 1/9 = 0.11111.References
* Vysochanskij, D F & Petunin, Y I (1980) Justification of the 3σ rule for unimodal distributions, "Theory of Probability and Mathematical Statistics" vol. 21 pp25-36
* [http://m.njit.edu/CAMS/Technical_Reports/CAMS02_03/report4.pdf Report (on cancer diagnosis) by Petunin and others stating theorem in English]
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