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Mixed Distributions

  1. Sep 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Let X ~ Poission ([tex]\lambda \theta[/tex]). Suppose [tex]\lambda[/tex] ~ gamma (h,h-1) and [tex]\theta[/tex] ~ generalized Pareto ([tex]\alpha[/tex], h-1,k). Show that the marginal distribution of X is

    [tex]\frac{\Gamma(\alpha + k) \Gamma(\alpha + h) \Gamma(\alpha + x) \Gamma(k + x)}{\Gamma(\alpha) \Gamma(h) \Gamma(k) \Gamma(\alpha + h + k + x)x!}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    My prof didnt really explain it.

    From what I've gathered in the book it's:

    [tex]\int^{\infty}_{- \infty} d \lambda d \theta[/tex] of the distribution of X, lambda, and theta.
  2. jcsd
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