Chebyshev's inequality

(ˈtʃɛbɪˌʃɒfs)

(Statistics) statistics the fundamental theorem that the probability that a random variable differs from its mean by more than k standard deviations is less than or equal to 1/k²

[named after P. L. Chebyshev (1821–94), Russian mathematician]

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Next there exists a finite constant K > 0 such that condition (i) holds for all f [member of] L Applying Chebyshev's inequality to the Lebesgue measure of the set [E.sub.f] = {x [member of] R | [(log(1 + [absolute value of ([f.sup.*](x))]).sup.p] > K/[delta]} for f [member of] L, the following estimate is valid:

Bounded Subsets of Smirnov and Privalov Classes on the Upper Half Plane

What is interesting about (2) in Theorem 8 is the possibility of combining it with Chebyshev's inequality to obtain rates of convergence.

Sample dependence in the maximum entropy solution to the generalized moment problem

ESTIMATION FOR UPPER BOUND OF CUMULATIVE PROBABILITY USING CHEBYSHEV'S INEQUALITY

Dependence of ESD charge voltage on humidity in data centers: part III--estimation of ESD-related risk in data centers using voltage level extrapolation and Chebyshev's inequality

It is stressed that Chebyshev's inequality is rarely used to set mean value confidence intervals (in our case fuel efficiency average for both data, experimental and calculated).

Application of Chebyshev's theorem for estimating C[O.sub.2] emissions due to overloading of heavy duty diesel trucks/Aplicacion del teorema de Chebyshev para estimar las emisiones de C[O.sub.2] por sobrecarga de las unidades de transporte terrestre de carga

Then by Chebyshev's inequality, we can obtain the conclusion easily.

Dynamics of a stochastic cooperative predator-prey system with Beddington-DeAngelis functional response

Using the Chebyshev's inequality to sin A/2, sin B/2, sin c/2 and cos A/2, cos c/2 we get

A new refinement of the inequality [summation] sin A/2 [less than or equal to] [square root of 4r+r/2r]

The idea of the proof of Theorem 2.1 is as for the binary search tree [6, Theorem 2.1] to use the well-known Chebyshev inequality for proving (ii), (iii) and (iv); (i) is very easy to prove.(Recall that Chebyshev's inequality is a useful tool for proving that a random variable is sharply concentrated about its mean value.)