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

• Jan 25th 2010, 06:36 PM
ryzeg
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 ($\displaystyle \lambda$)
$\displaystyle \lambda$ : exponential ($\displaystyle \theta$)

And I simply need the marginal distribution of X and the conditional density for $\displaystyle \lambda$ 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?
• Jan 26th 2010, 05:00 PM
statmajor
That's called a mixture of distributions where

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

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

http://www.mathhelpforum.com/math-he...ributions.html
• Jan 27th 2010, 08:22 PM
ryzeg
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?

$\displaystyle f_X(x) = \int^{\lambda=\infty}_{\lambda=0} \frac{\lambda^{x}}{x!} e^{-\lambda} \times \theta e^{-\theta x} d \lambda$
• Jan 27th 2010, 11:39 PM
Moo
Hello,

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

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 (Surprised)

What d'you think ? I would be pleased if I'm wrong !
• Jan 28th 2010, 06:44 AM
statmajor
Quote:

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.
• Jan 28th 2010, 06:46 AM
ryzeg
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.)
• Jan 28th 2010, 07:08 AM
ryzeg
It is impossible to evaluate the integral above, so that is no bueno.
• Jan 28th 2010, 07:17 AM
statmajor
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
• Jan 28th 2010, 07:25 AM
ryzeg
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,...
• Jan 28th 2010, 07:26 AM
ryzeg
Yeah, I started typing that before I read your last post :)

Thanks, man.