## The Annals of Probability

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

### A Note on an Inequality Involving the Normal Distribution

#### 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 in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994428

**Digital Object Identifier**

doi:10.1214/aop/1176994428

**Mathematical Reviews number (MathSciNet)**

MR614640

**Zentralblatt MATH identifier**

0457.60014

**JSTOR**

links.jstor.org

**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. Ann. Probab. 9 (1981), no. 3, 533--535. doi:10.1214/aop/1176994428. https://projecteuclid.org/euclid.aop/1176994428