## Prior and Posterior Distributions

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 $\theta$.