# Joint Gamma prior, Gamma likelihood with data; posterior distribution is...?

• March 30th 2008, 12:44 PM
sickofants
Joint Gamma prior, Gamma likelihood with data; posterior distribution is...?
(I'm afraid I'm not familiar with the syntax for the math tags. I could only include the pdf because I lifted it from wikipedia. I hope this isn't too offputting)
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I really need help with this. As I'm not at all hot on Bayesian inference and I just don't know where to go:
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Question
Independent prior distributions are given to two parameters; alpha and theta, where theta=alpha/beta. Thus alpha and beta (=alpha*theta) are not independent in the prior:

alpha~gamma(1, 0.4) [this is an exp(0.4) dist., right? does this help me?]
&
theta~gamma(4, 400).

Assume that given the values of alpha and beta, service times (s) are independent and have a gamma(alpha, beta) distribution. That is:
$f(s;\alpha,\beta) = s^{\alpha-1} \frac{\beta^{\alpha} \, e^{-\beta\,s} }{\Gamma(\alpha)} \ \mathrm{for}\ s > 0 \,\!.$
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Given 50 observations which sum to 9925 seconds,
(a) Find the joint posterior distribution of alpha and theta
(b) Find the marginal posterior densities of alpha and theta [I would just integrate out each parameter from the joint pdf, right?]
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I hope that it doesn't seem too cheeky asking a lengthy question for my first post - I just don't know where to start. I hope you'll take me for my word when I say that I intend to stick around and answer questions in areas where I am more adept.