Let X be a random variable following the n-dimensional Gaussian distribution with mean vector μ and covariance matrix Σ. The probability density function is denoted by f(x). It is a well-known fact that when the above pdf is integrated over the whole Euclidean space R

^{n}the result is 1. I would like to find out what's happening when the gaussian pdf is integrated over the half-space of R

^{n}that is defined by the hyperplane H: x

^{T}w+b=0, i.e., the region Ω = {x \in R

^{n}| x

^{T}w+b > 0}. Someone could observe that if the bias term, b, is equal to zero, i.e., the hyperplane passes through the origin, then - by symmetry - the integral should be equal to 1/2 (am I wrong?).

I would appreciate every single hint or helpful comment!

Thanks in advance guys!