# What is the probability of having at least X values greater than 0 given an n-dimensional multivariate normal distribution?

My Problem

Within the context of an n-dimensional multivariate normal distribution with estimates for the means and covariance matrix, I am trying to calculate the probability of 1 or more, 2 or more, 3 or more, etc. of the included dimensions having a score of 0 or greater. At the moment, I accomplish this through simulation. Specifically, I draw 10,000 samples from a multivariate distribution using the estimated parameters for the mean and covariance matrix and then calculate how many of the dimensions have at least 1, 2, 3, …, n values of 0 or greater for each data point. I can then calculate a proportion for each. This works, but it is cumbersome given my application.

My Question

Given that I am dealing with a multivariate normal distribution, I assume there must be an analytic solution that could calculate these probabilities without the need for simulation. However, I am having trouble finding a formula to do so. Therefore, I was hoping someone might tell me:

1. Is there a general purpose formula to calculate the probability of having at least X values greater than 0 given an n-dimensional multivariate normal distribution?
2. To address a related question, is there also a general purpose formula to calculate the probability that an arbitrary subset of the dimensions (e.g., Dimensions 1, 3 and 8) will each have a score of 0 or greater – ignoring the remaining dimensions?
• There are general formulas but except in special circumstances they won't do you much good. The special circumstances are where all the marginal means are zero and the variables are independent or where there are only two variables: then the formulas can be evaluated. Otherwise I believe you need to perform numerical integration. – whuber Dec 29 '16 at 19:14
• @whuber Not what I was hoping for ;) In my application, there are 8 variables, the means are around -2.0 and the variables are moderately correlated. So it sounds as though the formulae would not work. Am I correct that you would therefore recommend I stick with my simulation approach? Or would numerical integration be superior? Otherwise, if you convert your comment into an answer and include a brief explanation (or link) to the referenced formulae (in case it might be of use to others in the future) I will accept it. – jmfawcet Dec 30 '16 at 16:59