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This paper presents explicit formulae giving tight upper and lower bounds on the expectations of alpha-unimodal random variables having a known range and given set of moments. Such bounds can be useful in ordering of random variables in terms of risk and in PERT analysis where there is only incomplete stochastic information concerning the variables under investigation. Explicit closed form solutions are also given involving alpha-unimodal random variables having a known mean for two particularly important measures of risk—the squared distance or variance, and the absolute deviation. In addition, optimal tight bounds are given for the probability of ruin in the collective risk model when the severity distribution has an alpha-unimodal distribution with known moments.

Best Bounds on Measures of Risk and Probability of Ruin for Alpha Unimodal Random Variables When There Is Limited Moment Information