## Finding the marginal distribution of a random variable w/ a random variable parameter

I am a little shaky on my probability, so bear with me if this is a dumb question...

Anyway, these two random variables are given:

X : Poisson ($$\lambda$$)
$$\lambda$$ : Exponential ($$\theta$$)

And I simply need the marginal distribution of X and the conditional density for $$\lambda$$ given a value for X

I have all the equations for dependent distributions, but do not know how to apply them to this ostensibly easy problem...

Any help?

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age
 You'll need Bayes' rule for this. What results have you got so far?
 I was doing this, but I think it is wrong: $$f_X(x) = \int^{\lambda=\infty}_{\lambda=0} \frac{\lambda^{x}}{x!} e^{-\lambda} \times \theta e^{-\theta \lambda} d \lambda$$ Plugging this integral into Mathematica gives a really nasty output with a incomplete gamma function, and my TI-89T cannot evaluate it.

## Finding the marginal distribution of a random variable w/ a random variable parameter

I take that back; the integral is doable with a little manipulation. Damn machines...