I got this far:
now I'm stuck. Not sure how to integrate with respect to lambda. Any help would be greately appreciated.
I'm not sure what you're using for a generalized Pareto.
There are conflicting definitions.
Likewise there are two definitions of the Gamma.
I know what the Poisson is, but not the Poison (joke).
This Pareto is weird lloking to me, mine usually starts at 1, not 0.