Bound on the joint distribution of three real random variables with given two dimensional marginals Let $X,Y,Z$ be real r.v. with $(X,Y)$, $(Y,Z)$ and $(Z,X)$ centered unit normal. How large can $\mathbb E (XYZ)$ be?
 A: My entry is upper bound of $\sqrt{\frac 2 {\pi}}$ via $E(XYZ) \le E \frac { X^2|Y| + Z^2|Y|} 2 = E(X^2)E(|Y|)$ by the marginal independence of X and Y, and you can achieve $(\sqrt{\frac 2 {\pi}})^3$ by choosing  $Z = sgn(X) sgn(Y) |W|$, W independent of everything, which works as e.g. conditional on X having some positive value the distribution of Z is the same as $sgn(Y) |W|$, which is easily seen to be normal.
A: For a discretized version of this problem, $E(XYZ)$ can be as high as $5/(4\sqrt{3})$, or ~ 0.72.
Suppose that $X$, $Y$, $Z$, are all constrained to have values in $\{\pm \sqrt{3}, \pm 1 / \sqrt{3}\}$.  Let
\begin{align}
P(X=x, Y=y, Z=z) =
& 1/64 \text{ in the }\text{ 4 cases with } xyz = \sqrt{27}\\
& \ \ 0 \ \ \ \text{ in the 12 cases with } xyz = \sqrt{3}\\
& 3/64 \text{ in the 12 cases with } xyz = 1/\sqrt{3}\\
& 6/64 \text{ in the }\text{ 4 cases with } xyz = 1/\sqrt{27}\\
& \ \ 0 \ \ \ \text{ in the 32 cases with } xyz < 0\\
\end{align}
which reduces the calculation of $E(XYZ)$ to multiplying the numbers on each row and adding up the results.  The marginal distributions are
\begin{align}
P(X=x, Y=y) =
& 1/64 \text{ in the 4 cases with } xy = \pm 3\\
& 3/64 \text{ in the 8 cases with } xy = \pm 1\\
& 9/64 \text{ in the 4 cases with } xy = \pm 1/3\\
\end{align}
\begin{align}
P(X=x) =
& 1/8 \text{ in the 2 cases with } x = \pm \sqrt{3}\\
& 3/8 \text{ in the 2 cases with } x = \pm 1/\sqrt{3}\\
\end{align}
So each of $X,Y,Z$ is a variable with mean 0 and variance 1, and they are pairwise independent.
I expect that there are similar continuous distributions with similar values for $E(XYZ)$.
