# A problem about Multivariate Normal Distribution

$$(X,Y,Z)$$ 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 $$E[X^2YZ] = E[X^2]E[Y]E[Z]$$

• Do you have the covariance matrix? Nov 25, 2022 at 16:32
• You also need to know the means. The general solution for a bivariate Normal is given at stats.stackexchange.com/questions/583124/…. 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, 2022 at 16:40
• 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 Nov 25, 2022 at 19:10
• 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, 2022 at 14:20