Hi. I have this problem.
Suppose that the time in minutes required to servce a customer at a certain shop has an exponential distribution expressed as , , where the parameter is unknown. Suppose that the prior distribution of gamma with mean and standard 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 .