**What would be the distribution (p.d.f.) of the following ratio?**

$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$

where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can be seen, the denominator follows a Chi-square distribution with $2M$ degrees of freedom as $x_{i}$ are i.i.d. R.V.s. 

**Remark 1**: I've run some simulations in Matlab, as shown in the pictures below for $M$ = 10, and the resulting distribution has a bell-shaped histogram. Could it be that the resulting distribution follows one of these distributions: Gaussian/Cauchy/Student's-t?

**Remark 2**: This is a link to the Matlab/Octave script used to plot the pictures below. [Matlab/Octave simulation of the histogram of z][1]

[![Histogram of the Real part of z][2]][2]

[![Histogram of the Imaginary part of z][3]][3]

  [1]: https://pastebin.com/B8zVtDeK
  [2]: https://i.sstatic.net/3Eaqc.png
  [3]: https://i.sstatic.net/A2PHB.png