This paper presents an output feedback control law, hereafter called the Linear Quadratic Random Delay Compensator (LQRDC), for application to processes that are subjected to randomly varying distributed delays. LQRDC is synthesized in the stochastic setting based on dynamic programming. The closed-loop LQRDC system includes the plant dynamics, state estimator and delayed control commands, and is constructed in the sensor-time frame which may be time-skewed relative to the controller-time frame. A pair of discrete-time modified matrix Riccati equations and a pair of matrix Lyapunov equations are constructed by using Lagrangian multipliers and the matrix minimum principle. The performance cost is formulated as the conditional expectation of a quadratic functional adjoined with an equality constraint involving the dynamic of the conditional covariance of the closed-loop system state.