I am trying to compute multivariate normal distributions at some points.
I am using Matlab's
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 $2sigma^2$ from the mean and the density is almost the same?
Shouldn't the result be close to zero?
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.
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