Consider a multivariate Gaussian measure $$ d lambda (x): = nu _ { mu, Sigma} e ^ {- langle (x- mu), Sigma ^ {-1) (x- mu) rangle – vert x vert ^ 2} $$

with vector $ mu in mathbb R ^ n $ Y $ Sigma $ positive definite and $ nu _ { mu, Sigma} $ a normalization constant to convert $ d lambda $ in a measure of probability.

Leave $ m $ be the expected value of the vector $ m: = int _ { mathbb R ^ n} x d lambda (x). $

Then we consider the expected value for $ X $ distributed according to the measure $ lambda: $

$$ mathbb E left ( langle X-m, Sigma ^ {- 1} and rangle ^ 2 langle X-m, Sigma ^ {- 1} mu rangle right). $$

Question: Can we say something about the sign of this expected value for general vectors? $ y in mathbb R ^ n $? – How I got this expression, I guess this expression is never strictly positive, but I can't see it immediately.