You have a sequence of coins \(X_1,…,X_n\) drawn iid from a Bernouli distribution with unknown parameter \(p\) and known fixed \(n\). Assume a priori that the coins parameter \(p\) follows a Beta distribution with parameters \(\alpha,\beta\).

- Given the sequence \(X_1,…,X_n\) what is the posterior pdf of \(p\) ?
- For what value of \(p\) is the maximum of the posterior pdf attained.

Hint: If \(X\) is distributed Bernoulli(p) then for \(x=1,0\) one has \(P(X=x)=p^x(1-p)^{(1-x)}\). Furthermore, if \(X_1,X_2\) are i.i.d. Bernoulli(p) then

\[P(X_1=x_1, X_2=x_2 )=P(X_1=x_1)P(X_2=x_2 )=p^{x_1}(1-p)^{(1-x_1)}p^{x_2}(1-p)^{(1-x_2)}\]