$(X,Y,Z)$ is a multivariate normal distribution. Calculate $E[X^2YZ]$
I'm finding an approach for this problem. I'm not sure if it is possible to assume $E[X^2YZ] = E[X^2]E[Y]E[Z]$
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Do you have the covariance matrix? – utobi Nov 25 '22 at 16:32
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You also need to know the means. The general solution for a bivariate Normal is given at https://stats.stackexchange.com/questions/583124/how-to-compute-this-moment-of-a-bivariate-normal-distribution/583152#583152. The technique obviously extends to more variables. Alternatively, since $X^2YZ=X^2((Y+Z)/2)^2-X^2((Y-Z)/2)^2$ and linear combinations of multivariate Normals are also multivariate Normal, you can apply the bivariate solution separately to the bivariate Normal variables $(X, (Y+Z)/2)$ and $(X,(Y-Z)/2).$ If either $\mu_Y$ or $\mu_Z$ is zero, the answer trivially is zero. – whuber Nov 25 '22 at 16:40
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It only mentions that $(X,Y,Z)$ is a Gaussian vector,which I understand it's a multivariate normal distribution. Covariance matrix isn't given. So maybe there is an error with this problem – anormalguy Nov 25 '22 at 19:10
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There's not necessarily an error in the problem statement: it just means the answer will depend on the nine parameters of the distribution. – whuber Nov 26 '22 at 14:20