Bayesian estimation with given prior Given the i.i.d. random variables $X_{1}, X_{2},...,X_{n}$ with frequenty function: $f(x|p)=(x-1)(1-p)^{x-2}p^2$,  with $ x =2,3,...$
Pick as a priori density $f_{P}(p) \propto p(1-p)1_{(0,1]}(p)$.
Give the posterior estimate for $p$ based on this prior.
I find Bayesian estimation very hard to grasp, can someone help me out on this one?
 A: Outline and Hints: According to the applicable version of Bayes' Theorem, you have
$$f(p\,|\,\mathbf{x}) \propto f(p) \times f(\mathsf{x}\,|\, p).$$
Here $f(p\,|\,\mathbf{x})$ represents the posterior distribution of $p$ given data,
$f(\mathsf{x}\,|\, p)$ is the likelihood function of the data, and $f(p)$
is the PDF of the prior distribution [in your case $\mathsf{Beta}(2,2)].$
With your prior and data consisting of $x$ successes in $n$ binomial
trials, one can see that the posterior distribution would be $\mathsf{Beta}(2+x,\; 2+n-x).$ [This is easy to see looking at the kernel of the likelihood because the beta prior and binomial likelihood are 'conjugate' (mathematically compatible).]
You don't say what kind of Bayesian estimate you want. For point
estimates, the mode and mean (and even the median) of the posterior
distribution have been used. For a Bayesian 95% probability interval estimate
one method is to find quantiles .025 and .975 of the
posterior distribution. (Something like the qbeta function in R
can be used to do the numerical integration, if necessary.)
Addendum per comment: I answered for binomial data. But your likelihood function is $\prod_{i=1}^n f(x_i|p)$, for the given $f(x|p).$ Then you would get somewhat different beta parameters for the posterior. My guess is that the answer is the mode of the posterior distribution, but I don't have the full context of what you've been studying. In meetings next few days so can't quickly go further into it. If you can't get it on your own, then I suggest you 'unaccept' my 'outline and hints' and maybe someone else will help.
