The homogenization of elastic periodic plates is as follows: The 3D heterogeneous body is replaced by a homogeneous Love-Kirchhoff plate whose stiffness constants are computed by solving an auxiliary boundary problem on a 3D unit cell that generates the plate by periodicity in the in-plane directions. In the present study, a generalization of the above mentioned approach is presented for the random case. The homogenized bending stiffness and the moduli for in-plane deformation of a plate cut from a block of composite material, considered to be a statistically uniform random material in the in-plane directions, are defined in three equivalent manners: a) the first definition considers statistically invariant stress and strain fields in the infinite plate. In the second and third definitions, a finite representative volume element of the plate is submitted to suitable b) kinematically uniform boundary conditions and c) statically uniform boundary conditions. The relationships between these three definitions are studied and bounds are derived.