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Let $X = (X_1, ..., X_p)$ a random variable with a $N(\mu, \Sigma)$ distribution.

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$$ \Pr(X_1 > a_1, ..., X_p > a_p) \\ =\int_{a_1}^\infty ... \int_{a_p}^\infty (2\pi)^{-p/2} (\det(\Sigma))^{-1/2} \exp\left\{-\frac{1}{2} (x-\mu)' \Sigma^{-1} (x-\mu)\right\} dx_1 ... dx_p $$

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Does it exist an (analytical?) approximation of this multivariate integral?

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Not really an answer, but to know more about this I would take a look at the references given for the pmvnorm function of the mvtnorm R package

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