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Chebyshev Inequalities for Unimodal Distributions

Thomas M. Sellke and Sarah H. Sellke
The American Statistician
Vol. 51, No. 1 (Feb., 1997), pp. 34-40
DOI: 10.2307/2684690
Stable URL: http://www.jstor.org/stable/2684690
Page Count: 7
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Chebyshev Inequalities for Unimodal Distributions
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Abstract

Let g be an even function on R that is nondecreasing on [0, ∞), and let k be a positive constant. For random variables X that are unimodal with mode 0, and for random variables X that are unimodal with an unspecified mode, we derive sharp upper bounds on P(|X| ≥ k) in terms of Eg(X). The proofs consist largely of drawing a chord and a few tangent lines on graphs of cdf's.

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