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Generalized gauss-chebyshev inequalities for unimodal distributions

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Abstract

Letg be an even function on ℝ which is nondecreasing in |x|. Letk be a positive constant. Sharp inequalities relatingP(|X|≥k) toEg(X) are obtained for random variablesX which are unimodal with mode 0, and for random variablesX which are unimodal with unspecified mode. The bounds in the mode 0 case generalize an inequality due to Gauss (1823), whereg(x)=x 2. The bounds in the second case generalize inequalities of Vysochanskiĭ and Petunin (1980, 1983) and Dharmadhikari and Joag-dev (1985).

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  1. Department of Statistics, Mathematical Sciences Building, Purdue University, 47907-1399, West Lafayette, IN, USA

    Thomas Sellke

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  1. Thomas Sellke
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Sellke, T. Generalized gauss-chebyshev inequalities for unimodal distributions. Metrika 43, 107–121 (1996). https://doi.org/10.1007/BF02613901

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