If I have $N$ random variables (denoted below as $X_i$) with uniform distribution on the $x$-axis $X_i = \rm{rand}[0,1]$ then the sum
$$ S_N = \frac{1}{N}\sum_i^N\frac{1}{2X_i-1} $$
seems to be a Cauchy distribution with $\gamma = \pi/2$. I found this by trial and error. $$ f_{S_N}(x)=\frac2{4x^2 + \pi^2} $$
The plot of the distribution along with the ideal Cauchy distribution is shown below for $N=200$, with data accumulated from $10^5$ trials.
Cauchy Distribution http://www.rearviewminor.com/cauchydist2.png
However, what I would really like to find is the probability distribution of
$$ T_N = \frac{1}{N}\sum_i^N\frac{1}{X_i} $$
with the same flat random variable distribution $X_i = \rm{rand}[0,1]$ as above. For different values of $N$ (the distribution is no longer independent of $N$ like the first distribution) the distributions of $T_N-\log N$ are plotted, showing a convergence for large $N$.
Some Other Distribution http://www.rearviewminor.com/unknowndistshift2.png
All of the distributions seem to have right tails that go as $x^{-2}$, which makes sense in light of A.S.'s comment. Does anyone recognize what distribution this might be? Or better yet, teach a man to fish: how does one go from a sum of random variables to figuring out what the probability distribution is?
So far I have been doing some reading around on Wikipedia, and I found this pdf helpful, but I am still stuck. Solving this will ideally be a stepping stone on my road to finding an answer to my other question.