# Thread: Mixture of Gamma, Poison, and Pareto Distribution

1. ## Mixture of Gamma, Poison, and Pareto Distribution

I got this far:

$\frac{\Gamma(\alpha + k)}{\Gamma(h) \Gamma(\alpha) \Gamma(k)} \int^{\infty}_0 \int^{\infty}_0 \frac{e^{-(h^{-1} + \lambda) \theta}\theta^{h+x-1} \lambda^{k+x-1} h^{-2k}}{(1+h^{-1} \lambda)^{\alpha + k}}d \lambda d \theta$

now I'm stuck. Not sure how to integrate with respect to lambda. Any help would be greately appreciated.

2. 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.

3. $
f(X|\lambda\vartheta) = \frac{(\lambda\vartheta)^{x}e^{-\lambda\vartheta}}{x!}
$

$
g(\theta) = \frac{(h^{-1})^{h}}{\Gamma(h)}\theta^{h - 1}e^{-(h^{-1})\theta}
$

$h(\lambda) = \frac{\Gamma(\alpha + k)h^{-k} \lambda^{k-1}}{\Gamma( \alpha) \Gamma(k)(1+h^{-1}x)^{\alpha + k}}$

then I did this:

$
f_X(x) = \int^{\infty}_0 \int^{\infty}_0f(X|\lambda\vartheta) g(\theta) h(\lambda) d \lambda d\theta
$