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Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from the mean (0) in terms of d?

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This is a chi distribution with $n$ degrees of freedom, so called because it is the square root of a chi-square distribution.

Its density for $x\gt 0$ is $$\frac{2^{1-n/2}x^{n-1}e^{-x^2/2}}{\Gamma(n/2)}.$$

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