Suppose I have three uncorrelated random variables $X, Y$ and $Z$ (discrete or continuous) such that
$\newcommand{\Cov}{\mathrm{Cov}}\Cov(X,Y)=0;\quad \Cov(Y,Z)=0;\quad \Cov(X,Z)=0 \tag{$\ast$}$
I am to prove or disprove the fact that
$E(XYZ) = E(X)\cdot E(Y)\cdot E(Z)$
It is evening here already and my head is somewhat dumb; now the obvious step I took was that
$\Cov(XY, Z) = E(XYZ) - E(XY)\cdot E(Z)$
Now that $X$ and $Y$ are uncorrelated, it implies that
$\Cov(XY, Z) = E(XYZ) - E(X)\cdot E(Y)\cdot E(Z)$
If I could prove or disprove that from $(\ast)$ implies $\Cov(XY,Z)=0$, then I would succeed.
Can you please help me with that?
Thank you in advance!