Period-doubling bifurcation in a double-well Duffing-van der Pol system with bounded random parameters and subject to harmonic excitations is studied. The random system is reduced to its equivalent deterministic one by the Chebyshev polynomial approximation, through which the response of the random system can be obtained by deterministic numerical methods. Numerical simulations show that similar to their counterparts in deterministic nonlinear systems, period-doubling bifurcation may occur in the random Duffing-van der Pol system, and that the period-doubling bifurcation of the random_parameter system has its own characteristics. Numerical results also show that the Chebyshev polynomial approximation is an effective approach in solving dynamical problems of nonlinear systems with random parameters.