Stochastic solutions to the demand response (DR) problem enable utility companies to address the uncertainties in customer demand. In another words, stochastic DR solutions are recently developed considering storage systems, renewable resources, and the electricity market. However, a DR approach for uncertain response of consumers to an incentive-based DR problem has not been studied before. This article presents a new prospective stochastic DR approach by using a mixed-strategy Stackelberg game (MSG) between an aggregator and residential customers during peak load periods. We use real dataset collected by a local utility. We cluster customers based on customer baseline load (CBL), select one reference customer for each cluster, and model the MSG between that customer and the aggregator. We map the MSG to subgames and prove the equilibrium using mixed-strategy Nash Equilibrium. Our results show that the peak load can be shifted to off-peak hours for almost 40$\%$ of load, and players gain monetary rewards in return.