R Multivariate Normal CDF I am trying to estimate the CDF of a MVN distribution for some given ranges ([-Inf,mu], [mu,mu+3], [mu+3,Inf]). In R using package mvtnorm and function pmvnorm, with some dummy data:
mu=c(13,15,12)
co=matrix(c(6,2,3,2,4,2,3,2,5),nrow=3,byrow=T)

library(mvtnorm)
pmvnorm(lower=-Inf,
        upper=mu,
        mean=mu,
        sigma=co)[1]

pmvnorm(lower=mu,
        upper=mu+3,
        mean=mu,
        sigma=co)[1]

pmvnorm(lower=mu+3,
        upper=Inf,
        mean=mu,
        sigma=co)[1]

[1] 0.2415246
[1] 0.1123291
[1] 0.01065227

Clearly this does not sum to 1, even though I went from -Inf to Inf. Why not?
Also, if I try to estimate one of the ranges as the differences of the current minus the previous, for ex. for the last interval in [mu+3,Inf] = [-Inf,Inf] - [-Inf,mu+3]:
pmvnorm(lower=-Inf,upper=Inf,mean=mu,sigma=co)[1]-
pmvnorm(lower=-Inf,upper=mu+3,mean=mu,sigma=co)[1]

[1] 0.2010412

The result is different from the previous estimate. Why?
 A: The first pmvnorm  calculates the probability that variable 1 <=13 AND variable 2 <=15 AND variable <=12 all occurs at the same time. The probability that each individual variable fufills that criteria will be 0.5, however the joint probability will not be 0.5
If we use an example where all variables are uncorrelated
mu=c(13,15,12)
co=matrix(c(1,0.0,0,0,1,0,0,0,1),nrow=3,byrow=T)

pmvnorm(lower=-Inf,
        upper=mu,
        mean=mu,
        sigma = co)

This returns
[1] 0.125

This is the probability comes from the probability of three uncorrelated events happening at the same time, i.e. 0.5^3 = 0.125. In your data the variables seems to be correlated, which is the reason that probability is greater than 0.125. 
Basically the reason that the sum of probabilities does not equal one is that the calculations above doesn't include all possible coordinates. For example the coordinates could be (10, 17,14), which isn't included.
As for the difference in the estimate, I believe that the pmvnorm use some sort of sampling algorithm with may lead to differences in probability.
