Moment Generating Function w/ Condtional Expectation

[cse63146]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

So, this is what I interpreted the problem as:

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

e^xt = e^{$$\alpha \lambda$$t}

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

or would I have to do this:

e^xt = E[E[e^xt|$$\theta$$]]

 

[lippyka]

= E[e\alpha \lambdat]= e\alpha \lambda \int_0^{\infty} \frac{\alpha^{\alpha}}{\Gamma(\alpha)} t^{\alpha - 1} e^{-\lambda t} dt = e\alpha \lambda \frac{\alpha^{\alpha}}{\Gamma(\alpha)} \int_0^{\infty} t^{\alpha - 1} e^{-\lambda t} dt = e\alpha \lambda \frac{\alpha^{\alpha}}{\Gamma(\alpha)} \frac{\Gamma(\alpha)}{\lambda^{\alpha}}