Suppose that X1, . . . , Xn form a random sample from a distribution for which the p.d.f. f (x|θ) is as follows:

Suppose also that the value of the parameter θ is unknown (θ > 0), and the prior distribution of θ is the gamma distribution with parameters α and β (α > 0 and β > 0). Determine the mean and the variance of the posterior distribution of θ.

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f(x16) =16x6-1 for 0