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Given a multivariate normal $X \sim N(\vec{0},\Sigma)$, I would like to calculate the pdf when sampling from the unit ball $(||X||_2=1)$. Specifically what is the value of the normalizing factor $Z$ ie. the integral of the gaussian over the ball.

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    *Ball* $\to$ *sphere*.2012-12-05

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By change of variables, this is the same as computing the integral of the density of a standard normal $N(0,I)$ over an ellipsoid whose principal axes can be assumed to be the coordinate axes. The answer can be expressed in terms of elliptic integrals. AFAIK such a problem is often solved with a Monte Carlo simulation, if the dimension is high.