Binomial proportion confidence interval with known distribution of the proportion If I have a coin that produces head with probability p, I know there are several methods to estimate a confidence interval for p with a given confidence level.
If I also know that the coin was randomly selected from a set of coins whose distribution of p is known, I think that it is possible to improve the quality of the estimation.
I tried to find if this is a known problem and if there are known solutions, but I found nothing.
Is there anyone who knows if this is a studied problem or has any idea on how it could be possible to leverage the known distribution of p in order to improve its estimation?
 A: I don't think either of the comments under the question quite get at what's going on here. 
My understanding of it is this (and please correct me if I am wrong): a specific value for $p$ is drawn from a known distribution, but we don't observe that $p$. Instead we observe a binomial random variable with that $p$. You want to use both the information in the known distribution for $p$ as well as in the likelihood to make some inference about $p$. 
On that understanding:
A random $p$ obtained from a known distribution plus some information about that specific draw of $p$ from a sample is naturally answered with  Bayesian approach; the known distribution of $p$ is the prior, which you then update via the sample (specifically the posterior is proportional to the product of the likelihood and the prior).
$$f(p|X)\,\propto\, f(X|p)\,f(p)$$
For example, if the prior were beta, the posterior would also be beta (since the 
beta is conjugate for a binomial likelihood). 
If the prior were symmetric triangular, the posterior would be piecewise beta (with two continuous pieces).
More generally we could use a variety of techniques to obtain the updated information about $p$. On this simple problem - assuming a prior that didn't work nicely with the likelihood - one option would use numerical integration to scale the product of the prior and the likelihood to a density.
A: Thanks everybody for the help. 
I am indeed in an easy situation, where I can approximate my prior distribution with a beta distribution 
Beta($\alpha_0$ ,$\beta_0$). 
Therefore, the posterior distribution is 
Beta($\alpha_0$ + observed_heads , $\beta_0$ + observed_tails).
I found an answer to another question that uses a very nice example that can be applied to my case: https://stats.stackexchange.com/a/47782/44624
