[Math] need help to get $Cov(XY,Z)$

Question

I have three variables X, Y, Z. and I know $Corr(X,Y), Corr(X,Z),Corr(Y,Z),Var(X),Var(Y),Var(Z)$. and Cov is Covariance, Corr is correlation, Var is variance, $Corr(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)*Var(Y)}}$ Can I get $Cov(XY,Z)$

Answer 1

You do not seem to have enough information. It can be shown (I used a developmental version of mathStatica here to automate, but one could do this manually) that:

$$\text{Cov}(XY, Z) = \mathbb{E}[X] \text{Cov}(Y,Z)+\mathbb{E}[Y] \text{Cov}(X,Z)+\mu _{1,1,1}$$

where $$\mu _{1,1,1}=\mathbb{E}\left[\;(X-\mathbb{E}[X]) \; (Y-\mathbb{E}[Y]) \;(Z-\mathbb{E}[Z]) \; \right]$$

You know all the bivariate covariances, but you would still need to know the raw means $\mathbb{E}[X]$, $\mathbb{E}[Y]$ and the product central moment $\mu _{1,1,1}$ or ... the product raw moment $\mathbb{E}[X Y Z]$ and $\mathbb{E}[X]$, $\mathbb{E}[Y]$, $\mathbb{E}[Z]$.