Chebyshev Inequality


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Related to Chebyshev Inequality: variance, Markov inequality
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Chebyshev Inequality

 

(1) A basic inequality for monotonic sequences or functions. In the case of the finite sequences a1a2 ≤ . . . ≤ an and blb2 = ≤ . . . ≤ bn it has the form

In integral form we have

Here, f(x) ≥ 0 and g(x) ≥ 0; in addition, the functions are either both increasing or both decreasing. The inequality was derived by P. L. Chebyshev in 1882.

(2) An inequality that provides an estimate of the probability that the deviation of a random variable from its mathematical expectation exceeds some given limit. Let ξ be a random variable, Eξ = a its mathematical expectation, and Dξ = σ2 its variance. The Chebyshev inequality asserts that the probability of the inequality

|ξ – a| ≥ kσ

does not exceed the quantity 1/k2. If ξ is the sum of independent random variables and some additional restrictions are made, then the estimate 1/k2 can be replaced by the estimate 2 exp (–k2/4), which decreases with increasing k much more rapidly.

The inequality is named for P. L. Chebyshev, who used it in 1867 to establish extremely broad conditions for the application of the law of large numbers to sums of independent random variables. (SeeLARGE NUMBERS, LAW OF; LIMIT THEOREMS.)

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Mo, "Chebyshev inequality with estimated mean and variance," The American Statistician, vol.38, no.2, pp.130-132, 1984.
Its proof is based on a similar result by Kellogg for homogeneous polynomials and the univariate Chebyshev inequality. It was shown by Kellogg [2] (see also [4]) that for every homogeneous polynomial of degree m given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Chebyshev inequality tells us that the probability of being within 2 standard deviations is at least .75.