What is the radius of a "Gaussian" sphere such that approx. all the population lie within?Let X∈BRn be a random vector distributed normally, i.e. X∼N(0,σ2In), where σ&gt;0 is the standard deviation and In is the identity matrix of order n.SNKIn the case of the univariate Gaussian distribution, i.e.,X∼N(0,σ2), 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., X∼N(0,σ2In)? That is, what would be the radius of the n-sphere in terms of the standard deviation σ, 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.

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What is the radius of a &quot;Gaussian&quot; sphere such that approx. all the population lie within?Let X∈BRn be a random vector distributed normally, i.e. X∼N(0,σ2In), where σ&amp;gt;0 is the standard deviation and In is the identity matrix of order n.SNKIn the case of the univariate Gaussian distribution, i.e.,X∼N(0,σ2), 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 &quot;isotropic&quot; Gaussian distribution, i.e., X∼N(0,σ2In)? That is, what would be the radius of the n-sphere in terms of the standard deviation σ, 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.

Porter Camacho · Accepted Answer

The distribution of |X|2σ2 is χk2. Since for k=1.P(|x|≤3σ)=.9973002I interpret this as the question for which r in the case of k=3P(|X|≤rσ)=.9973002.This is r=F−1(.997302)=3.76205 where F is the cumulative distribution function of a χ32 random variable.

What is the radius of a "Gaussian" sphere

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