In this paper, we develop a new tube-based robust economic MPC scheme for linear time-invariant systems subject to bounded disturbances with given distributions. By using the error distribution in the predictions of the finite horizon optimal control problem, we can incorporate stochastic information in order to improve the expected performance while being able to guarantee strict feasibility. For this new framework, we can provide bounds on the asymptotic average performance of the closed-loop system. Moreover, a constructive approach is presented in order to find an appropriate terminal cost leading to a slight degradation of the bound on the guaranteed average performance. (C) 2016 Elsevier Ltd. All rights reserved.