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

  1. Jan 25, 2010 #1
    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?
     
  2. jcsd
  3. Jan 27, 2010 #2
    Re: Finding the marginal distribution of a random variable w/ a random variable param

    You'll need Bayes' rule for this. What results have you got so far?
     
  4. Jan 28, 2010 #3
    Re: Finding the marginal distribution of a random variable w/ a random variable param

    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.
     
  5. Jan 28, 2010 #4
    Re: Finding the marginal distribution of a random variable w/ a random variable param

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