I'm having some issues with the following task - It would be perfect if somebody could help me, please. I really would like to understand the details which means that I need really detailed explanations step by step (especially the contex and the math behind):

Let ₁, ₂, ..., ₁₀ be a random sample from a gamma distribution with = 3 and = 1/. Suppose we believe that follows a gamma-distribution with = 17 and = 72 and suppose we have a trial (₁ , ... , ) with an observed = 39.29

a) Find the posterior distribution of .

b) What is the Bayes point estimate of associated with the square-error loss function?

c) What is the Bayes point estimate of using the mode of the posterior distribution?