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

[ryzeg]

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?

 

[bpet]

You'll need Bayes' rule for this. What results have you got so far?

 

[ryzeg]

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.

 

[ryzeg]

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