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This paper proposes a Stackelberg game approach to deal with demand response (DR) scheduling under load uncertainty based on real-time pricing (RTP) in a residential setup. We formulate the optimization problems for the service provider that acts as the leader and for the users who are the multiple followers, respectively. We derive the Stackelberg Equilibrium (SE) consisting of the optimal real-time electricity price and each user's optimal power consumption. Simulation results show the proposed DR scheduling not only can control the total power consumption, but also is beneficial to the service provider's revenue. Moreover, the load uncertainty increases the service provider's revenue and decreases each user's payoff.