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 context 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?

I would be highly appreciated if you could do the calculations very detailed as this is my biggest problem do understand the steps...