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This paper deals with a new filtering problem for linear uncertain discrete-time stochastic systems with randomly varying sensor delay. The norm-bounded parameter uncertainties enter into the system matrix of the state space model. The system measurements are subject to randomly varying sensor delays, which often occur in information transmissions through networks. The problem addressed is the design of a linear filter such that, for all admissible parameter uncertainties and all probabilistic sensor delays, the error state of the filtering process is mean square bounded, and the steady-state variance of the estimation error for each state is not more than the individual prescribed upper bound. We show that the filtering problem under consideration can effectively be solved if there are positive definite solutions to a couple of algebraic Riccati-like inequalities or linear matrix inequalities. We also characterize the set of desired robust filters in terms of some free parameters. An illustrative numerical example is used to demonstrate the usefulness and flexibility of the proposed design approach.