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 f(x;\theta)=\theta exp(-\theta x), x \ge 0, where the parameter \theta is unknown. Suppose that the prior distribution of gamma with mean 0.2 and standard deviation 0.1. The average time of service for a random sample of 20 customers is obserced to be 3.8 minutes.

a) Find an expression for the joint density of the individual customer service times X_1, ... ,X_20.
So, the answer is just \Pi_{n=1}^{20}f(x_i;\theta_1, ... , \theta_20)

b) Find an expression for the prior density of \theta.
I'm not quite sure. I worked out Gamma's k value is 4 and \theta=0.05, 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 \theta.

e) Use the mean of this posterior to estmate \theta.