  We consider $N\times N$ Hermitian or symmetric random matrices with
independent entries. The distribution of the $(i,j)$-th matrix element is given
by a probability measure $\nu_{ij}$ whose first two moments coincide with those
of the corresponding Gaussian ensemble. We prove that the joint probability
distribution of the components of eigenvectors associated with eigenvalues
close to the spectral edge agrees with that of the corresponding Gaussian
ensemble. For eigenvectors associated with bulk eigenvalues, the same
conclusion holds provided the first four moments of the distribution $\nu_{ij}$
coincide with those of the corresponding Gaussian ensemble. More generally, we
prove that the joint eigenvector-eigenvalue distributions near the spectral
edge of two generalized Wigner ensembles agree, provided that the first two
moments of the entries match and that one of the ensembles satisfies a level
repulsion estimate. If in addition the first four moments match then this
result holds also in the bulk.
