 Solution

For a binomial distribution, the score function isIts squared expected value is $I(\theta )=n/[\theta (1-\theta )]$, using the binomial distribution variance $n\theta (1-\theta )$. Hence the Jeffreys prior isthe Beta distribution $\mathrm{Beta}(1/2,1/2)$. The factor $\sqrt{n}$ is independent of $\theta$ and disappears on normalization.