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 ([tex]\lambda[/tex])
[tex]\lambda[/tex] : Exponential ([tex]\theta[/tex])

And I simply need the marginal distribution of X and the conditional density for [tex]\lambda[/tex] 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:

[tex]
f_X(x) = \int^{\lambda=\infty}_{\lambda=0} \frac{\lambda^{x}}{x!} e^{-\lambda} \times \theta e^{-\theta \lambda} d \lambda
[/tex]

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...