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.
Publié le : 1981-06-14
Classification:
Inequality,
normal distribution,
Hermite polynomials,
isoperimetric problem,
60E05,
26A84
@article{1176994428,
author = {Chernoff, Herman},
title = {A Note on an Inequality Involving the Normal Distribution},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 533-535},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994428}
}
Chernoff, Herman. A Note on an Inequality Involving the Normal Distribution. Ann. Probab., Tome 9 (1981) no. 6, pp. 533-535. http://gdmltest.u-ga.fr/item/1176994428/