The ground-up loss X for an insurance policy is modelled by a discrete mixture of Exp (1) and Exp (2) with survival function x>0,

where [0,1] is the mixing weight. Policy limit of level 4 and ordinary deductible of level 0.5 are applied to each loss. Suppose that the samples of the amount paid per payment were recorded with sample mean 1. Calculate the estimator of using the method of moments.

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