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For a general multivariate normally distributed $\boldsymbol{X}$, what is the expectation of $1/(\boldsymbol{X}^T \boldsymbol{X})$
You can use a second order Taylor approximation.
Let $Y = X^TX$ with $X\sim \mathcal{N}(\mu, \Sigma)$. Then what you want is $\mathbb{E}(f(Y))$, where $f(x)=\frac{1}{x}$. The second order Taylor ...
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