The design of a gear system involves the consideration of various random parameters. To ensure that vibrations occurring during operation are controlled, it is crucial to assess the sensitivity of the system to these factors. The objective is to study the effects of uncertain parameters on the dynamic response of a wind turbine's two-stage gearbox. To this end, a 12-degree-of-freedom lump-sum model is proposed, which takes into account the uncertainty associated with the system's mass, moment of inertia, bearing bending stiffness, tension-compression stiffness, damping coefficient, and wind speed. Next, the dynamic response of the two-stage gear system under uncertain design parameters is analyzed using polynomial chaos (PC) expansion with different orders as well as different standard deviations, which is confirmed by Monte Carlo simulation (MCS), containing 100,000 runs.