Inspired by Stein’s lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive $E\left[X_1^{2}f\left(\mathbf{X}\right)\right]$ for any measurable function $f$ satisfying some regularity conditions. Then, by making use of this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of $E\left[X_1^{2}f\left(\mathbf{X}\right)\right]$ for multivariate Student-$t$, logistic and Laplace distributions.

中文翻译：

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椭圆分布的联合矩的表达式

受斯坦因引理的启发，我们得出了椭圆分布联合矩的两个表达式。我们使用两种不同的方法来推导$Ë\left[X_{\mathrm{1个}}^{2}F（\mathbf{X}）\right]$ 对于任何可测量的功能 $F$满足一些规律性条件。然后，利用该结果，我们得到了对正态分布随机变量乘积的期望的新公式，并给出了$Ë\left[X_{\mathrm{1个}}^{2}F（\mathbf{X}）\right]$ 对于多元学生-$Ť$，物流和拉普拉斯分布。