Finding Joint Posterior Density of Poisson and Exp?


Some friends and I have been stuck on this bayesian stats question for five days now, please shed some light if you can :wave:
We've tried to multiply the the Poisson posterior density and exponential posterior density together to produce a joint density, but we're told it's not correct :/

The question:

You’re working for a motor insurer. You are asked to model a pool of policies.
You are told to:
- Assume the occurrence of claims is a Poisson process with rate λ
- The size of the claims (dollars) comes from an Exponential distribution Exp(θ).
- Claim sizes are independent of each other and independent of the time of the claims.
- Independent Gamma(0.2,0.2) prior distributions should be used for λ and θ.

You are given motor policy data for the past two years. The data says there were n=20 claims during the two years with a total value of y=100,000 USD. You need to use these data to estimate the posterior predictive probability p, that the total value of claims occurring during the next six months exceeds 80,000 USD.

QUESTION: Give an expression for the joint posterior density π(λ, θ|n, y). Provide a clear account of the processes of going from prior to posterior to predictive distributions.

Thanks a bunch >.<
Last edited: