# Multivariate normal density function

I am trying to compute multivariate normal distributions at some points.

I am using Matlab's mvnpdf function: y = mvnpdf(X,MU,SIGMA)

The first argument is the point where I compute the density, MU is the mean and SIGMA the covariance.

I am puzzled by the following result:

 mvnpdf([0 0 0],[0 0 0],0.001*eye(3))

ans =

2.0078e+03

>> mvnpdf([0 0 0.002],[0 0 0],0.001*eye(3))

ans =

2.0038e+03


I am going at $2\sigma^2$ from the mean and the density is almost the same? Shouldn't the result be close to zero?

• $\sigma \neq \sigma^2$ (except in special cases!) if you know what I mean. But, that is only part of the reason. :-) – cardinal Oct 30 '13 at 17:15

What you are specifying with the second argument to mvnpdf is, as you correctly state, the (co-) variance. The standard deviation corresponding to a variance of 0.001 is about 0.0316. The point you are looking at is at a distance of 0.002 from the center of the distribution, or about 0.0632 standard deviations from the center, i.e. it is very close to the center. It is therefore to be expected that the density is only slightly smaller than at the center.