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The distributed robust fusion Kalman filtering problem is investigated in this paper for uncertain stochastic systems with random observation delays and missing measurements. The existing results are generalized to the case where each sensor subsystem may fail or be delayed at any sample time independently of the others, the random delays and missing measurements are described by multiple Bernoulli random processes and their probabilities are assumed to be known. For robust performance, stochastic parameter perturbations are introduced in the system matrices. The local robust optimal filter is derived in the linear minimum variance sense by using the innovation analysis approach. Then, the estimation error cross-covariance matrix between any two sensor subsystems is derived. A distributed robust fusion Kalman filter is obtained based on the optimal fusion algorithm weighted by matrices in the linear minimum variance sense. The performance of the designed fusion filter is dependent on the measurement missing probabilities. Simulations for a tracking system with two sensors show the effectiveness of the proposed design.