Given a multivariate Gaussian over the domain, $\Bbb{R}^n$ with mean $\mu$ and covariance matrix, $\Sigma$. For simplicity sake, we can consider the case where $\Sigma$ is a diagonal matrix. I'd like to find the probability that this distribution lies in the region: $\sum c_i x_i \leq b$. We can assume the $c_i$'s and $b$ are all positive if it matters. In other words, the area under the curve for that region.
The original problem was for a multivariate Binomial consisting of $n$ independent binomials. The problem was that there were a lot of points in the region of interest and the probability had to be calculated repeatedly.
So, I figured approximating them with a multivariate Gaussian would make it simpler but if this can be done for a multivariate Binomial, even better.
One way to do this would be with simulation. Importance sampling will give us a pretty good estimate with a few samples. I'm wondering if there is something more exact. Perhaps with numerical integration?