  The halo occupation distribution (HOD) describes the bias between galaxies
and dark matter by specifying (a) the probability P(N|M) that a halo of virial
mass M contains N galaxies of a particular class and (b) the relative
distributions of galaxies and dark matter within halos. We calculate predicted
HODs for a Lambda-CDM cosmological model using an SPH hydrodynamic simulation
and a semi-analytic (SA) galaxy formation model. Although the two methods
predict different galaxy mass functions, their HOD predictions agree remarkably
well. For mass-selected samples, the mean occupation <N(M)> exhibits a sharp
cutoff at low halo masses, a slowly rising plateau for <N>~1-2, and a more
steeply rising high occupancy regime. At low <N>, the mean pair and triple
counts are well below Poisson expectations, with important consequences for
small scale behavior of 2- and 3-point correlation functions. The HOD depends
strongly on galaxy age, with high mass halos populated mainly by old galaxies
and low mass halos by young galaxies. The SPH simulation supports several
simplifying assumptions about HOD bias: the most massive galaxy in a halo
usually lies close to the center and moves near the halo's mean velocity;
satellite galaxies have the same radial profile and velocity dispersion as the
dark matter; and the mean occupation at fixed halo mass is independent of the
halo's larger scale environment. By applying the SPH and SA HODs to a large
volume N-body simulation, we show that both methods predict slight, observable
departures from a power-law galaxy correlation function. The predicted HODs are
closely tied to the underlying galaxy formation physics, they offer useful
guidance to theoretical models of galaxy clustering, and they will be tested
empirically by ongoing analyses of galaxy redshift surveys. (Shortened)
