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?
Thanks!!
The joint distribution of X and p is...
$\displaystyle f(x,p)=p(1-p)^{x-1}$
where 0<p<1 and x=1,2,3,...
In order to obtain f(p|X) you need the marginal of X.
$\displaystyle 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
$\displaystyle {\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.