I have three RVs $X$, $Y$ and $Z$, where $X,Y$ are pairwise dependent, but $X,Z$ are independent and $Y,Z$ are also independent.
I also know the expectation of $Z$ is zero.
Can I say anything about $\mathbb{E}[XYZ]$?
I would like to know whether Cov$(XY, Z) = 0$.
I know it's not true that $\mathbb{E}[XYZ]$ = $\mathbb{E}[X]\mathbb{E}[Y]\mathbb{E}[Z]$, but I don't know whether I can say that $\mathbb{E}[XYZ]$ = $\mathbb{E}[XY]\mathbb{E}[Z]$.
By hypothesis $E(Z)=0$. Hence
$$\operatorname{Cov}(XY,Z) = E(XYZ) - E(XY)E(Z) = E(XYZ).$$
As $X,Y$ may be mutually dependent, but are both independent from $Z$, the probability distribution function - pdf of $(X,Y,Z)$ is $h(x,y,z)=f(x,y)g(z)$ where $f(x,y)$ is the pdf of $(X,Y)$ and $g(z)$ the one of $Z$. Therefore
$$\begin{aligned}E(XYZ)&= \int\int\int xyz f(x,y) g(z) dx \ dy \ dz\\ &= \int \int \left(\int z g(z) dz \right) xy f(x,y) dx \ dy\\ &=0 \end{aligned}$$ and $\operatorname{Cov}(XY,Z) =0$ as well.