Chebyshev inequality

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for finite monotone sequences

The inequality

Chebyshev's inequality for monotone functions is the inequality

where and are either both increasing or both decreasing on .

The inequalities were established by P.L. Chebyshev in 1882.


It is not important that and be non-negative. The proof consists of simply integrating the non-negative function over the square .

How to Cite This Entry:
Chebyshev inequality. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article