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We revisit the recent experiments [Billy et al., Nature 453, 891 (2008); Roati et al., Nature 453, 895 (2008)] that have lead to the observation of Anderson localization for matter waves in one-dimensional speckle and quasiperiodic potentials. We show that in presence of localization the tails of the asymptotic density distribution can be described by the diagonal term of the projection over the eigenstates...