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June, 1981 A Note on an Inequality Involving the Normal Distribution
Herman Chernoff
Ann. Probab. 9(3): 533-535 (June, 1981). DOI: 10.1214/aop/1176994428


The following inequality is useful in studying a variation of the classical isoperimetric problem. Let $X$ be normally distributed with mean 0 and variance 1. If $g$ is absolutely continuous and $g(X)$ has finite variance, then $E \{\lbrack g'(X)\rbrack^2\} \geq \operatorname{Var}\lbrack g(X)\rbrack$ with equality if and only if $g(X)$ is linear in $X$. The proof involves expanding $g(X)$ in Hermite polynomials.


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Herman Chernoff. "A Note on an Inequality Involving the Normal Distribution." Ann. Probab. 9 (3) 533 - 535, June, 1981.


Published: June, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0457.60014
MathSciNet: MR614640
Digital Object Identifier: 10.1214/aop/1176994428

Primary: 60E05
Secondary: 26A84

Keywords: Hermite polynomials , inequality , isoperimetric problem , normal distribution

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 3 • June, 1981
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