  Consider $n$ independent random numbers with a uniform distribution on
$[0,1]$. The number of them that exceed their mean is shown to have an Eulerian
distribution, i.e., it is described by the Eulerian numbers. This is related
to, but distinct from, the well known fact that the integer part of the sum of
independent random numbers uniform on $[0,1]$ has an Eulerian distribution. One
motivation for this problem comes from voting theory.
