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Finding the conditional distribution

  1. Nov 20, 2014 #1
    Hey guys, I'm trying to find a conditional distribution based on the following information:

    ##Y|u Poisson(u \lambda)##, where ##u~Gamma( \phi)## and ##Y~NegBinomial(\frac{\lambda \phi}{1+ \lambda \phi}, \phi^{-1})##

    I want to find the conditional distribution ##u|Y##

    Here's what I've got so far:

    ##f(u|y)= \frac{f(u, y)}{p(y)} = \frac{p(y|u)}{p(y)} f(u)##
    ##=\frac{(u \lambda)^{y}e^{-u \lambda}}{y!} \frac{u^{ \alpha - 1}e^{-u/ \beta}}{ \Gamma ( \alpha) \beta^{ \alpha}} \frac{y! \Gamma ( \alpha)}{\Gamma (y+ \alpha)} ( \frac{1+\lambda \beta}{ \lambda \beta})^{y} (1+ \lambda \beta)^{ \alpha}##

    where ##\beta = \phi## and ## \alpha = \phi^{-1}##
    (I'm using ##\beta## and ##\alpha## right now because that's what it does in the notes)

    I'm not sure where to go from here. I can cancel some terms out, but then I'm not sure what distribution I'm supposed to end up with. Could anyone give me a push in the right direction?
     
  2. jcsd
  3. Nov 25, 2014 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Nov 26, 2014 #3

    haruspex

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    Then please do so and post what you get.
     
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