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In this work, we have performed Monte Carlo simulations to study a mixed spin-1 and spin-3/2 Ising ferrimagnetic system on a square lattice with two different random single-ion anisotropies. This lattice is divided in two interpenetrating sublattices with spins S A = 1 in the sublattice A and S B = 3 ∕ 2 in the sublattice B . The exchange interaction between the spins on the sublattices is antiferromagnetic ( J < 0 ). We used two random single-ion anisotropies, D i A and D j B , on the sublattices A and B , respectively. We have determined the phase diagram of the model in the critical temperature T c versus strength of the random single-ion anisotropy D plane and we shown that it exhibits only second-order phase transition lines. We also shown that this system displays compensation temperatures for some cases of the random single-ion distribution. [ABSTRACT FROM AUTHOR]