Let Weibull(k, λ) represent the distribution with pdf given by f(x) = λkx^(k−1) e^ (−λ(x)^k) 1(x > 0).
and consider the hierarchical model X1, . . . , Xn|λ ∼ Weibull(k, λ) λ ∼ Gamma(α, β), where k > 0 is known and α and β are chosen by the investigator.
(a) Identify the posterior density π(λ|X1, . . . , Xn) of λ given the data.
(b) Give an expression for the posterior mean of λ given the data.
(c) Treating λ as a fixed constant as in the frequentist paradigm. . .
i. Give the likelihood function for λ based on X1, . . . , Xn.
ii. Give the log-likelihood function for λ based on X1, . . . , Xn.
iii. Give the MLE of λ.
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