Abstract:

Let ($X_1$,$X_2$,$X_3$) be a centered Gaussian random vector with $D(X_i)=1$, $i=1,2,3$. By means of the properties of hypergeometric function and factorization, we prove that

$E\big[|X_1^4X_2^3X_3^3|\big]\geq$E$|X_1^4|$E$|X_2^3|$E$|X_3^3|$,

and the equal sign holds if and only if $X_1$,$X_2$,$X_3$ are independent. This complements the results of the three dimensional Gauss product inequality in the existing literature.

Key words: Gauss Product Inequality, normal Distribution, hypergeometric Function, factorization