We compute analytically, for large N, the probability distribution of the
number of positive eigenvalues (the index N_{+}) of a random NxN matrix
belonging to Gaussian orthogonal (\beta=1), unitary (\beta=2) or symplectic
(\beta=4) ensembles. The distribution of the fraction of positive eigenvalues
c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-\beta N^2 \Phi(c)] where
the rate function \Phi(c), symmetric around c=1/2 and universal (independent of
β), is calculated exactly. The distribution has non-Gaussian tails, but
even near its peak at c=1/2 it is not strictly Gaussian due to an unusual
logarithmic singularity in the rate function.Comment: 4 pages Revtex, 4 .eps figures include