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.
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:
- 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?
- 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?