This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.


Let $(X\_i:1\le i\le N)$ be a multivariate normal random vector where i) each coordinate $X\_i$ is standard normal and ii) $\mathbb{E}[X\_iX\_j]=\rho$ for every $1\le i<j\le N$.


My question. Is there a symbolic expression (as a function of $N$ and $\rho$ only) for the following probability:
$$
\mathbb{P}\left(X\_1\ge 0,X\_2\ge 0,\dots,X\_N\ge 0\right).
$$
Some further notes:


* For $N=2$, using the fact $X\_1$ and $\frac{X\_2-\rho X\_1}{\sqrt{1-\rho^2}}$ are i.i.d. standard normal, one can indeed reach to $\frac14+\frac{\sin^{-1}\rho}{2\pi}$.
* For $N=3$, such formulas are available in standard textbooks, and is given by $\frac18+\frac{3\sin^{-1}\rho}{4\pi}$.


Situation gets more involved beyond $N\ge 4$.


**Edit.** Looking at the structure of the solution for $N=2$ and $N=3$, I postulate that the answer is
$$
\frac{1}{2^N}+\binom{N}{2}\frac{\sin^{-1}\rho}{\eta}
$$
where
$$
\eta = \binom{N}{2}\pi \frac{2^{N-1}}{2^{N-1}-1}.
$$
My rationale is a) for $\rho=0$ it must be $2^{-N}$ as everybody is independent, b) the term, $\sin^{-1}\rho$, should appear exactly once for each pair $1\le i<j\le N$, and c) for $\rho=1$, it should be $\frac12$.


This obviously simplifies to
$$
2^{-N}+\frac{\sin^{-1}\rho\left(2^{N-1}-1\right)}{2^{N-1}\pi}
$$



On the probability of the multivariate normal with fixed pairwise correlations being coordinate-wise non-negative
E.g., [Ruben, formulas (102) and (102') on p. 220](https://www.jstor.org/stable/2333017?seq=1) has a recurrence for your probability
$$
p\_n:=P(X\_1\ge0,\dots,X\_n\ge 0).
$$
It is stated there, on p. 213: "For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer be expressed in terms of elementary functions."
See also e.g. [Plackett](https://www.jstor.org/stable/2332716?origin=crossref&seq=1) and references there for more general results.




---


Anyhow, your conjectured expression for $p\_n$ cannot be true in general. Indeed, the eigenvalues of your covariance matrix of $(X\_1,\dots,X\_n)$ are $1+(n-1)\rho$ (with the eigenspace spanned by the vector $\mathbf1:=(1,\dots,1)$) and $1-\rho$. Since the covariance matrix must be positive semidefinite, necessarily $1+(n-1)\rho\ge0$, that is, $\rho\ge-1/(n-1)$.


In particular, if $\rho=-1/(n-1)$, then the random point $(X\_1,\dots,X\_n)$ is with probability $1$ on the hyperplane passing through the origin and orthogonal to the vector $\mathbf1$. Therefore, then $p\_n=0$, which does not agree with your conjectured expression for $p\_n$.


