Three-dimensional hydrogen/air turbulent premixed flames with detailed chemistry are investigated using direct numerical simulation (DNS). A fourth-order explicit Runge-Kutta method for time integration and an eighth-order central differencing scheme for spatial discretization are used to solve the fully compressible Navier-Stokes (N-S) equation system. The simulations are carried out in a cubic box initialized with the corresponding one-dimensional laminar premixed flame. The boundaries are periodic in the lateral directions and non-reflecting in the direction perpendicular to the flame front. Three cases with different equivalence ratios of the fresh gas are investigated, all of which are within the thin reaction zone regime. The preferential diffusion effect in premixed combustion is examined. The presumed probability density functions (PDFs) for the progress variable are studied. Conditional moment closure (CMC) submodels are validated for the velocity, scalar dissipation rate, and reaction rate. It is found that the linear model captures the variations of the conditional axial velocity for all of the cases studied. The scalar dissipation rate is modeled using the amplitude mapping closure (AMC) model. However, the performance of the AMC model is not good for the premixed flames. The first-order approximation for the reaction rate is excellent.