Hi. I have this problem.

Suppose that the time in minutes required to servce a customer at a certain shop has anexponential distributionexpressed as , , where the parameter is unknown. Suppose that theprior distributionof gammawithmeanandstandard deviation. The average time of service for a random sample of customers is obserced to be minutes.

a) Find an expression for the joint density of the individual customer service times .

So, the answer is just

b) Find an expression for the prior density of .

I'm not quite sure. I worked out Gamma's k value is and , but I don't think there's anthing to do with that.

c) Determine the posterior distribution up to a constant of proportionality.

i don't know how to do this...

d) Hence deduce the posterior distribution of .

e) Use the mean of this posterior to estmate .