Assume we have a Multivariate Normal distribution. For simplicity, let all N random variables have a zero mean and unit variance. Also for simplicity, let the correlation between all pairs of random variables be a constant rho.
Question: what is the probability that in any particular realization, K out of the N random variables will have the same sign?
The natural extensions are to allow each r.v. to have its own, non-zero, mean, and to allow any positive definite covariance matrix, but even the simplest case seems difficult. Any assistance would be greatly appreciated.
Thanks in advance
Thomas Philips
Denote $f(x_1,\ldots,x_N) $ the density of zero mean Gaussian random vector vith covariance $\Sigma$ such that \begin{align} (\Sigma)_{ii}&=1 \\ (\Sigma)_{ij}&=\sigma. \end{align} Denote $P_{1,\ldots,K}$ the probability that $X_1,\ldots,X_K$ are negative and $X_{K+1},\ldots,X_N$ are positive $$ P_{1,\ldots,K} = P[X_1< 0,\ldots,X_K < 0,X_{K+1} > 0,\ldots, X_N >0], $$then $$ P_{1,\ldots,K}= \underbrace{\int_{-\infty}^0 \cdots \int_{-\infty}^0}_{K \text{ times }} \underbrace{\int_{0}^{\infty} \cdots \int_{0}^{\infty}}_{N-K \text{ times }} f(x_1,\ldots,x_N)d(x_1,\ldots,x_N). $$ It is obvious that $f$ is invariant under permutation of variables and therefore $$P_{1,\ldots,K}=P_{i_1,\ldots,i_K}$$ where $i_1,\ldots,i_K$ is any combination of indexes $\{1,\ldots,N\}$ of length $K$. Therefore the probability that exactly $K$ are negative is $$\binom{N}{K}P_{1,\ldots,K}.$$