Consider $x_i$ iid random variables with $E[x_i]=1$ and $x_i>0$. While I know we can't get a distribution for
\begin{equation} \frac{\sum_{i=1}^{n} x_i} {\sum_{i=1}^{n} x_i^2} \end{equation}
in general, is there any specific cases where this random variable has a well known distribution.
Some attempts and thoughts. For large $n$ the denominator and numerator are both approximately normal via the central limit theorem. There are a few papers on ratio distributions of normal random variables (when the means of the denominator and numerator are 0, and the denominator and numerator are independent, their ratio distribution is just the cauchy distribution - of course neither of those hassumptions are true here).
As a first guess I was thinking that some discrete distribution e.g. $x_i=3/2$ with probability $1/2$ and $x_i=1/2$ with probability $1/2$ might be a promising example where one might be able to generate a pmf for this quantity, but that quickly became a mess. I'm wondering if there is any case where the pmf or pdf of this ratio is analytically tractable?