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Linear programming formulations cannot handle the presence of uncertainty in the problem data and even small variations in the data can render an optimal solution infeasible. A number of robust linear optimization techniques produce formulations (not necessarily linear) that guarantee the feasibility of the optimal solutions for all realizations of the uncertain data. A recent robust approach in  maintains the linearity of the formulation and is able to strike a balance between the conservatism and quality of a solution by allowing less robust solutions. In this work we demonstrate how to use distributional information on problem data in robust linear optimization. We adopt the robust model of  and present an approach that exploits distributional information on problem data to decide the level of robustness of the formulation, thus, leading to much more cost-effective solutions (by 50% or more in some instances).We apply our methodology to a stochastic inventory control problem with quality of service constraints.