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

1. 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?

2. That's called a mixture of distributions where

$f(x) = \int^{\infty}_0 f(X| \lambda) f(\lambda) d \lambda$

where $f(X| \lambda) : Poisson (\lambda)$ and $f(\lambda ) : exponential (\theta)$

http://www.mathhelpforum.com/math-he...ributions.html

3. Thanks, that sounds good.

But I have an even dumber question -- how do I get rid of the thetas if I am only integrating out the lambdas? Are the thetas constants? Am I even setting the integral up right?

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

4. Hello,

The $\theta$ are treated as constants.
But you set your integral wrong : $f(\lambda)=\theta e^{-\theta {\color{red}\lambda}}$

But I'm not sure it's really a mixture... See the wikipedia articles : http://en.wikipedia.org/wiki/Mixture_model and http://en.wikipedia.org/wiki/Mixture_density for the definitions...
Also, I can't see the formula you've been given

What d'you think ? I would be pleased if I'm wrong !

5. Originally Posted by Moo
But I'm not sure it's really a mixture... See the wikipedia articles : Mixture model - Wikipedia, the free encyclopedia and Mixture density - Wikipedia, the free encyclopedia for the definitions...
Also, I can't see the formula you've been given
Maybe it's not. In my textbook, there's a section called "Mixture Distributions" and the questions/examples are similiar in nature.

6. That makes more sense.

"Also, I can't see the formula you've been given"

What formula? When I said, "I have all the equations for dependent distributions", I was referring to the fact that I have all the books and material required for this type of problem, but I just do not know which one to use.

"What d'you think ? I would be pleased if I'm wrong !"

I was thinking of using Bayes' theorem. Would that yield the same answer? (I guess I will find out... working on it now.)

7. It is impossible to evaluate the integral above, so that is no bueno.

8. In order to evaluate, you'll need to read up on the Gamma Function:

You'll be able to see it easier if you take out the constants from the integral.

http://en.wikipedia.org/wiki/Gamma_function

9. OKAY, I take that back. With a little manipulation, it is doable.

u = lambda(1+theta)
du=(1+theta)dlambda
lambda^x=u^x/(1+theta)^x

So you are left with the integral,

C*integral(u^k*e^(-u),u,0,inf)

Where C = theta/(x!(1+theta)^(x+1))

Which makes the answer theta/(1+theta)^(x+1) for x=0,1,...

10. Yeah, I started typing that before I read your last post

Thanks, man.