Expected value of $XYZ$, $E(XYZ)$, is not always a $E(X)E(Y)E(Z)$, even if $X$, $Y$, $Z$ are not correlated in pairs Could you prompt me, please, is it true?
Expected value of $XYZ$, $E(XYZ)$, is not always $E(X)E(Y)E(Z)$, even if $X$, $Y$, $Z$ are not correlated in pairs, because if $X$, $Y$, $Z$ are not correlated in pairs it doesn't entail that they are uncorrelated in aggregate (it is my idea)?

Let $\Omega=\{\omega_1,\omega_2, \omega_3, \omega_4\}$, $\mathcal{F}=2^{\Omega}$ and $\mathbb{P}(\{\omega_i\})=1/4$ for all $i$. It is easy to check that desired random variables are
$$
X=1_{\{\omega_1,\omega_2\}},\quad Y=1_{\{\omega_1,\omega_3\}},\quad Z=1_{\{\omega_1,\omega_4\}}
$$
