## Prior and posterior distributions

A geologist researching seismic activity in south-east Turkey is able to specify her prior beliefs regarding the parameter of the Poisson distribution. She tells us that $\displaystyle \theta$ can be modelled with a gamma distribution, i.e. $\displaystyle \theta\sim\Gamma(\alpha,\lambda)$, and so our prior distribution for $\displaystyle \theta$ is of the form

$\displaystyle \pi(\theta)=\frac{\lambda^{\alpha}}{\Gamma(\alpha) }\theta^{\alpha-1}e^{-\lambda\theta}, \theta, \alpha, \lambda\geq0$

Specifically the geologist specifies that $\displaystyle \theta\sim\Gamma(6,2)$

a) Use the prior distribution for $\displaystyle \theta$ specified by the geologist, and the likelihood function...

$\displaystyle L(\theta|\underline{x})=\frac{e^{-2\theta}\theta^9}{4320}$

...to obtain the posterior distribution for $\displaystyle \theta$ in light of this data $\displaystyle \pi(\theta|\underline{x}=(6,3))$

b) What is the poterior mean for the number of seismic earth tremors per week? How has the mean changed in light of the data?

I've gotten myself in the right tizzy with this. Now I know how to find the posterior, but adding this with extra data I'm not entirely sure what I'm meant to be doing. Any ideas?