  We study the distribution of occupation times for a one-dimensional random
walk restricted to a finite interval by reflecting boundary conditions. At
short times the classical bimodal distribution due to L\'evy is reproduced with
walkers staying mostly either left or right to the initial point. With
increasing time, however, the boundaries suppress large excursions from the
starting point, and the distribution becomes unimodal converging to a
$\delta$-distribution in the long time limit. An approximate spectral analysis
of the underlying Fokker-Planck equation yields results in excellent agreement
with numerical simulations.
