For a class of discrete-time nonlinear systems with state delays, input delays, and stochastic disturbances, this paper investigates the identification and predictive control of the output probability density function (PDF). To address the coexistence of state delays, input delays, and random disturbances, an output PDF evolution model is established based on an augmented-state description, and a parameterized representation satisfying nonnegativity and normalization is derived. Based on Optimal Transport theory, the Wasserstein distance is employed to construct an identification criterion for the output PDF, and a parameter update mechanism together with the corresponding identification error dynamics is developed, under which conditions for bounded convergence and asymptotic convergence of the identification error are obtained. On this basis, a model predictive controller for output PDF tracking is designed, in which the distribution tracking error, system constraints, and terminal stability requirements are incorporated into a unified optimization framework. Furthermore, by means of a Lyapunov-Krasovskii functional, it is proved that the closed-loop system is asymptotically stable in the distributional sense or uniformly ultimately bounded. The results show that the proposed method achieves an integrated design of identification and control for the output PDF of delayed nonlinear stochastic systems. Numerical simulations and data from a grinding-classification process further verify the effectiveness and practical applicability of the proposed approach. □