# Is there a formula for $\text{Cov}(XY, ZW)$?

I know that $X$ and $Y$ are independent as well as $Z$ and $W$. I am trying to find a formula for $\text{Cov}(XY,ZW)$. I started by writing it down as \begin{align}\text{Cov}(XY,ZW)&=E(XYZW)-E(XY)E(ZW)\\[2ex]&=E(XYZW)-E(X)E(Y)E(Z)E(W)\end{align} and I am stuck. I am not sure if I can write $E(XYZW)$ as $E(XZ)E(YW)$, and even if I can, I am not sure if that would be helpful.

• You can't and it wouldn't be. – Jake Westfall Mar 16 '17 at 14:21
• @Matthew Cant he? $X$ and $Y$ are independent. – Łukasz Grad Mar 16 '17 at 14:37
• @ŁukaszGrad My mistake: I read it too fast. if $X$ and $Y$ are independent then obviously $\operatorname{Cov}(X,Y) = 0$ and $\operatorname{E}[XY]=\operatorname{E}[X]\operatorname{E}[Y]$. That's fine. Continuing forward though, $\operatorname{E}[XYZW] = \operatorname{Cov}(XZ,YW) + \operatorname{E}[XZ]\operatorname{E}[YW]$ won't be particularly helpful. – Matthew Gunn Mar 16 '17 at 15:00
• @ŁukaszGrad The substitution requires not that $X$ be independent of $Y$, which is assumed, but that $XY$ be independent of $ZW$, which does not follow from the assumptions (at least not as I have read and understood them -- I understand them to be just $X \perp Y$ and $Z \perp W$.) Maybe the OP can clarify. – Jake Westfall Mar 16 '17 at 20:36

I'm afraid that knowing only $X \perp Y$ and $Z \perp W$ will not get you very far. The best you can do is probably:
$$\text{Cov}(XY,ZW) = \text{E}(X)\text{E}(Z)\text{Cov}(Y,W) + \text{E}(X)\text{E}(W)\text{Cov}(Y,Z) + \text{E}(Y)\text{E}(Z)\text{Cov}(X,W) + \text{E}(Y)\text{E}(W)\text{Cov}(X,Z) + \text{E}(X)\text{E}[\Delta_Y\Delta_Z\Delta_W] + \text{E}(Y)\text{E}[\Delta_X\Delta_Z\Delta_W] + \text{E}(Z)\text{E}[\Delta_X\Delta_Y\Delta_W] + \text{E}(W)\text{E}[\Delta_X\Delta_Y\Delta_Z] + \text{E}[\Delta_X\Delta_Y\Delta_Z\Delta_W]$$
where $\Delta_A = A - \text{E}[A]$.