Let $a_{1}, a_{2}\in\mathbb{R}^{d}$, $X\sim\mathcal{N}(0,\Sigma)$ be a Gaussian random vector in $\mathbb{R}^{d}$ and let $p$ denote its density. Consider the following integral \begin{equation} \int_{\{x:~ a_{1}^{\top}x>0,~ a_{2}^{\top}x>0\}}p(x)dx, \end{equation} which is the measure (under $p$) of the intersection of the halfspaces determined by $a_{1}$ and $a_{2}$. Is this integral known in closed form?
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If $X$ is your $d$-dimensional random vector, the pair $(U,V)=(a_1^TX, a_2^TX)$ has a joint $2$-dimensional Gaussian distribution, from which it is easy to work out your integral. – kimchi lover Mar 24 '19 at 15:22
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@kimchilover Sure. I am actually trying to compute the joint pmf of the vector $(U',~V')$, wher $U'=\text{sign}(U)$ and $V'=\text{sign}(V)$. To compute $P(U'=1, ~V'=1)$, I need to evaluate the integral above – nemo Mar 24 '19 at 15:33
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As @kimchilover pointed out, the joint density $p_{UV}$ of $(U,V)$ can be found, and this integral is basically integration of $p_{UV}$ over the positive orthant and this is discussed here https://math.stackexchange.com/questions/869502/multivariate-gaussian-integral-over-positive-reals. But for higher dimensions, there doesn't seem to be a closed form expression – nemo Mar 24 '19 at 16:19