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Math Help - posterior distribution

  1. #1
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    posterior distribution

    P(X=k) = p(1-p)^k-1

    if p have a uniform prior distribution on [0,1], what is the posterior distritubion of p? and what is the posterior mean?


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  2. #2
    MHF Contributor matheagle's Avatar
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    The joint distribution of X and p is...

    f(x,p)=p(1-p)^{x-1}

    where 0<p<1 and x=1,2,3,...

    In order to obtain f(p|X) you need the marginal of X.

    P(X=x) =\int_0^1 p(1-p)^{x-1}dp=\int_0^1 p^{2-1}(1-p)^{x-1}dp

    NOW use the beta density http://en.wikipedia.org/wiki/Beta_distribution

     {\Gamma (2)\Gamma (x)\over \Gamma (x+2)} = {(x-1)!\over (x+1)!}= {1\over x(x+1)}

    where x=1,2,3,...

    This is a telescoping series that does sum to one.

    Now divide and you should have that conditional distribution.
    Last edited by matheagle; October 25th 2009 at 07:24 PM.
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