The Annals of Probability

A Note on an Inequality Involving the Normal Distribution

Herman Chernoff

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Abstract

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.

Article information

Source
Ann. Probab. Volume 9, Number 3 (1981), 533-535.

Dates
First available: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176994428

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176994428

Mathematical Reviews number (MathSciNet)
MR614640

Zentralblatt MATH identifier
0457.60014

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 26A84

Keywords
Inequality normal distribution Hermite polynomials isoperimetric problem

Citation

Chernoff, Herman. A Note on an Inequality Involving the Normal Distribution. The Annals of Probability 9 (1981), no. 3, 533--535. doi:10.1214/aop/1176994428. http://projecteuclid.org/euclid.aop/1176994428.


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