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)?

Suppose
$$
(X,Y,Z) = \begin{cases}
(1,1,0) & \text{with probability }1/4 \\
(1,0,1) & \text{with probability }1/4 \\
(0,1,1) & \text{with probability }1/4 \\
(0,0,0) & \text{with probability }1/4
\end{cases}
$$
Then $X,Y,Z$ are pairwise independent, and $E(X)E(Y)E(Z)=1/8\ne 0 = E(XYZ)$.
