  Inspired by Stein's lemma, we derive two expressions for the joint moments of
elliptical distributions. We use two different methods to derive
$E[X_{1}^{2}f(\mathbf{X})]$ for any measurable function $f$ satisfying some
regularity conditions. Then, by applying this result, we obtain new formulae
for expectations of product of normally distributed random variables, and also
present simplified expressions of $E[X_{1}^{2}f(\mathbf{X})]$ for multivariate
Student-$t$, logistic and Laplace distributions.
