  Oriented closed curves on an orientable surface with boundary are described
up to continuous deformation by reduced cyclic words in the generators of the
fundamental group and their inverses. By self-intersection number one means the
minimum number of transversal self-intersection points of representatives of
the class. We prove that if a class is chosen at random from among all classes
of $m$ letters, then for large $m$ the distribution of the self-intersection
number approaches the Gaussian distribution.
