## About Trivariate Normal Distribution

I am working on a paper, and have a problem regarding ellipsoid and Trivariate Normal Distribution. Suprisingly I can’t find much in literature but I found your in one of your answers:

Because this construction has nothing to do with “confidence” per se,

the objective is to establish some convention for describing the shape

and relative size of the points. Using 1.96 sort of works (for three

variables): it contains about 72% of the probability of the trivariate

normal distribution. But as the number of variables increases this

method produces ellipses that are far too small. For instance, with 10

variables it will contain only 4.6% of the probability; using 4.28

instead of 1.96 in this case will contain 95% of the probability.

How did you get this number 72%? Or do you have some literature to recommend to me in which I can find this. I would appreciate it very much!

If $X sim N_k(mu,Sigma)$, then $Q=(X-mu)’Sigma^{-1}(X-mu) sim chi^2_k$. Further, the level sets of $Q$ are the ellipsoids you refer to. So the 72% you mention comes from a chi-square distribution (these calcs in R):

As do the other numbers:

See http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Prediction_Interval