That's called a mixture of distributions where
where and
http://www.mathhelpforum.com/math-he...ributions.html
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 ( )
: exponential ( )
And I simply need the marginal distribution of X and the conditional density for 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?
That's called a mixture of distributions where
where and
http://www.mathhelpforum.com/math-he...ributions.html
Hello,
The are treated as constants.
But you set your integral wrong :
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 !
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.)
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
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,...