# pr.probability – Expected value sign

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.