Synchronization is an essential factor of multiplex dynamic networks. We study synchronization among multiplex delay networks with random switching nonlinear coupling under random disturbances. Different from the existing research works, this paper focuses on multiplex delay nonlinear coupling networks with Bernoulli random switching topologies. To overcome the time-varying dynamic characteristics of the multiplex networks and describe the limitation in signal transmission process, an adaptive delay-dependent quantitative tracking control policy is presented. To achieve synchronization among the multiplex switched delay networks in a limited time, finite time control technology is introduced into the design of the adaptive quantitative controller in the policy. The theoretical framework about the stochastic synchronization among the multiplex switched nonlinear coupling networks is derived. We find that the setting time is related to quantization level, parameters of controllers and Lyapunov functions. Finally, an example is presented to show the effective for the theoretical framework.