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I would like to compute analytically the following expected value:

$E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right)$

where the $X_i \approx N(0,1)$ are iid.

It seems an elementary integral, but it is eluding me. Any pointer to a non-trivial solution technique\technique, or the solution itself of course, is highly appreciated.

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# Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value:

$E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right)$

where the $X_i \approx N(0,1)$ are iid.

It seems an elementary integral, but it is eluding me. Any pointer to a non-trivial solution technique\, or the solution itself of course, is highly appreciated.