What is the radius of a "Gaussian" sphere such that approx. all the population lie within?
Let be a random vector distributed normally, i.e. , where is the standard deviation and is the identity matrix of order n.SNKIn the case of the univariate Gaussian distribution, i.e.,
, we say that approximately 99.7% of the population lie within three standard deviations.
What would be an appropriate constant in the case of the multivariate "isotropic" Gaussian distribution, i.e., ? That is, what would be the radius of the n-sphere in terms of the standard deviation , such that approximately all the population lie within?
P.s.: Please feel free to edit the title (and/or the body of this question). I was not sure how to express what I wanted in the title.