A numerical model is formulated for the nonlinear random vibrations of the offshore floating structures moored by cables under seismic and sea wave excitations. The upper end of each mooring cable is connected to the floating structure and the other end of each cable is fixed to the seabed. Nonlinear equations of motions of the mooring cables are formulated by using the nonlinear cable elements that are formulated based on the extended Hamilton principle. The floating platform is modeled as a rigid body with three degrees of freedom. The connection conditions which represent the relationships between the displacements of the floating platform and cables are given to derive the equations of motions of the whole system. The effects of nonlinear hydrodynamic drag forces and added-mass on both the floating platform and cables are taken into consideration. The random vibrations of the moored floating structure under both horizontal seismic ground motion and sea wave excitation are analyzed by using the Monte Carlo simulation method. The probability density functions of the displacements of the moored floating structure and the maximum tensile force in the cables are analyzed. The influences of different sag-to-span ratios, inclination angles and diameters of the mooring cables on the mean value and standard deviation of the displacements of the floating structure and the maximum tensile force in the cables are studied.