Top: Uniform, Bottom: ?? Distribution. Ignore the random spikes - those are just binning errors.
Looking for a distribution that is on $[0,1]$ and is equal to $0$ at $1$ and some positive constant at $0$. The distribution starts off flat (horizontal?) and then tapers off to zero.
Before I just succumb to making it up myself with some arbitrary function, I wanted to check if such a distribution or similar already exists in common mathematical literature.
EDIT: I played around with the data and got clean CDFs,
This would be Fermi-Dirac distribution: \begin{equation} f(\epsilon)=\frac{1}{e^{(\epsilon-u)/T}+1} \end{equation}
Theory in short:
imagine we have levels $\epsilon_i$. Each level $\epsilon_i$ has $g_i$ sublevels and can be occupied with $n_i<g_i$ indistinguishable particles such that no more than one particle resides on one sublevel. We are given total number of particles $N=\sum_i n_i$ and total energy $E=\sum_i n_i \epsilon_i$. Average occupation of each level $\epsilon_i$ is given by $\bar{n}(\epsilon_i) = \frac{\bar{n}_i}{g_i}$. Occupation that corresponds to maximal number of ways to populate levels $\epsilon_i$ given constraints on $N$, $E$, is Fermi-Dirac distribution $f(\epsilon)$. In the formula $u$ and $T$ corresponds to $N$ and $E$ respectively.
see also: http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics