Mathematics Homework Help

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.