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Let $[x_1, x_2, x_3]$ are three independent points in Cartesian space which are Gaussian distributed with a non-zero mean and identity covariance.

I need to calculate the following expectations,

\begin{align} \mathrm{E}\left[ \frac{x_i}{\sqrt{x_1^2 + x_2^2 + x_3^2}}\right],\quad \mathrm{E}\left[ \frac{x_i^2}{\sqrt{x_1^2 + x_2^2 + x_3^2}}\right],\quad \mathrm{E}\left[ \frac{x_ix_j}{\sqrt{x_1^2 + x_2^2 + x_3^2}}\right] \end{align}

I know the denominator is a non-central $\chi$-distributed variable with three degrees of freedom. But couldn't solve by plugging in the pdfs. I think the individual ratios should be non-central T distributed but how to get the values is the problem !

Thank you for your help.

Nikhil Sharma
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    The sum of the squares is not Chisquared (as the title suggests): given non-zero means, the sum of squares is non-central Chisquared (and presumably not independent of the numerator). – wolfies Nov 03 '22 at 06:11
  • @wolfies, You are right, it's non-central Chi distribution. Also, the denominator and numerator are dependent. – Nikhil Sharma Nov 03 '22 at 14:47
  • It's not what you are asking for, because the denominators in my problem are chi-squared, not chi, but maybe the pointers mentioned in this other answer can be a useful starting point to look for references or approximations https://stats.stackexchange.com/questions/605559/expected-value-of-the-outer-product-of-normalized-non-centered-gaussian-vector – dherrera Mar 15 '23 at 20:01

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