et X1, . . . , Xn ∼ iid Gamma(3,1/θ) and we assume the prior for θ is InvGamma(10,2). (a) Find the posterior distribution for θ. (b) If n = 10 and ¯x = 18.2, find the Bayes estimate under squared error loss. (c) The variance of the data distribution is ϕ = 3θ2. Find the Bayes estimator (under squared error loss) for ϕ.

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Let X1, . . . , Xn ∼ iid Gamma(3,1/θ) and we assume the prior for θ is InvGamma(10,2).
(a) Find the posterior distribution for θ.
(b) If n = 10 and ¯x = 18.2, find the Bayes estimate under squared error loss.
(c) The variance of the data distribution is ϕ = 3θ2. Find the Bayes estimator (under squared error loss) for ϕ. 

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Please explain how you did likelihood in part a. How did you get the θ3n and also the -θ/x for the exp?

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