# Mixed Distributions

1. Sep 25, 2009

### cse63146

1. The problem statement, all variables and given/known data

Let X ~ Poission ($$\lambda \theta$$). Suppose $$\lambda$$ ~ gamma (h,h-1) and $$\theta$$ ~ generalized Pareto ($$\alpha$$, h-1,k). Show that the marginal distribution of X is

$$\frac{\Gamma(\alpha + k) \Gamma(\alpha + h) \Gamma(\alpha + x) \Gamma(k + x)}{\Gamma(\alpha) \Gamma(h) \Gamma(k) \Gamma(\alpha + h + k + x)x!}$$

2. Relevant equations

3. The attempt at a solution

My prof didnt really explain it.

From what I've gathered in the book it's:

$$\int^{\infty}_{- \infty} d \lambda d \theta$$ of the distribution of X, lambda, and theta.