I wanted find the following expression in closed-form
$\mathbb{E} [sgn(X_1) sgn(X_2) sgn(X_3) sgn(X_4)]$
where each $X_i$ is a gaussian random variable correlated with others. In this regard, we have to find the probability of each possible case $\mathbb{E} [sgn(X_1) sgn(X_2) sgn(X_3) sgn(X_4)] = \mathbb P(X_1>0, X_2>0, X_3>0, X_4>0) - \mathbb P(X_1<0, X_2>0, X_3>0, X_4>0) + \ldots$
here we have the closed form for $\mathbb P(X_1>0, X_2>0, X_3>0, X_4>0)$ based on this Multivariate gaussian integral over positive reals. but unfortunately not for other terms.
Please guide if you can. Thanks for your help in advance.
