# Moment Generating Function w/ Condtional Expectation

1. Aug 15, 2009

### cse63146

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

Suppose $$\theta$$ ~ $$~ gamma(\alpha , \lambda)$$ where alpha is a positive integer. Conditional on $$\theta$$, X has a Poission distribution with mean $$\theta$$. Find the unconditional distribution of X by finding it's MGT.

2. Relevant equations

3. The attempt at a solution

So, this is what I interpreted the problem as:

X = E[E[X|$$\theta$$]] = E[$$\theta$$] = $$\alpha \lambda$$

ext = e$$\alpha \lambda$$t

It's as far as I got. Any hints?

or would I have to do this:

ext = E[E[ext|$$\theta$$]]

Last edited: Aug 15, 2009