I have a mixture of multivariate normal distributions, and I want to compute the integral with the first element of the input vector varying between specified limits, and the other elements varying from -infinity to +infinity. See attached pdf for equations. I've done it numerically but would like an analytic solution. Any advice? Thanks!

 

And here's the pdf.intMix2.pdf

I have a mixture of multivariate normal distributions, and I want to compute the integral with the first element of the input vector varying between specified limits, and the other elements varying from -infinity to +infinity. See attached pdf for equations. I've done it numerically but would like an analytic solution. Any advice? Thanks!

 

And here's the pdf.intMix2.pdf

 

The gaussian is integrable in closed form from -infinity to + infinity (using polar coordinates and a "trick") but there is in general no closed form expression for the integral of a gaussian over a finite interval.

 

You are stuck with numerical approximations or,what is equivalent, with tables.