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Let $(X_1, X_2, X_3)\sim N(\mu,\Sigma)$ be a three-dimensional random variable where each coordinates are dependent (i.e. $\Sigma$ has non-zero values outside of its diagonal)

I want to know how to compute

  1. $E(X_1 X_2^2)$

  2. $E(X_1 X_2 X_3)$

Thanks.

EDIT : I think I know how to do it. Using Isserlis Theorem, we know that $E((X_1-\mu_1) (X_2-\mu_2) (X_3-\mu_3))=0$. Also, using the identity $E(X_i X_j) = E(X_i)E(X_j) + Cov(X_i,X_j)$

I expand the expression $E((X_1-\mu_1) (X_2-\mu_2) (X_3-\mu_3))=0$ and use the covariance identity to get:

$$E(X_1 X_2 X_3)=E(X_1) E(X_2) E(X_3) +E(X_1)Cov(X_2,X_3) +E(X_2)Cov(X_1,X_3) +E(X_3)Cov(X_1,X_2)$$

or put differently $$E(X_1 X_2 X_3)=\mu_1\mu_2\mu_3 +\mu_1\sigma_{23} +\mu_2\sigma_{13} +\mu_3\sigma_{12}$$

To get $E(X_1 X_2^2)$, Can I use the derived result and say that $X_3=X_2$ all the time ? I simply replace the "3"s by 2 in the formula to get

$$E(X_1 X_2^2)=E(X_1 X_2 X_2)=\mu_1\mu_2\mu_2 +\mu_1\sigma_{22} +\mu_2\sigma_{12} +\mu_2\sigma_{12} =\mu_1(\mu_2^2 +\sigma_2^2) +2\mu_2\sigma_{12} $$

Can somebody confirm that I am not doing something wrong ?

Thank you.

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