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A Note on an Inequality Involving the Normal Distribution

Herman Chernoff
The Annals of Probability
Vol. 9, No. 3 (Jun., 1981), pp. 533-535
Stable URL: http://www.jstor.org/stable/2243541
Page Count: 3
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A Note on an Inequality Involving the Normal Distribution
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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.

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