I was trying to compute the expectation of something where something of the form $$\mathbb{E}\left[\frac{\sum_i X_i}{\sum_i X_i^2}\right]$$ appears multiple times, where $X_i \sim \mathcal{N}(\mu_x, \sigma_x^2)$ is normally distributed.
Now, I have this feeling that this form should be something that could be known in statistics. After all this looks very much like the quotient of empirical mean and (uncorrected) variance, but I don't seem to find anything on what this could be distributed like or what this expectation is.
I am aware that this form looks very similar to the t-distribution, but since the step from variance to standard deviation is non-linear, I am not sure whether this relation would help me in any way.
Is there anybody out there who could give me some pointer(s) what distribution this is or at least how I could compute the expectation of something like this?