= Solution
For a <binomial distribution>, the <score function> is
$$
\frac{Y}{\theta}-\frac{n-Y}{1-\theta}
=\frac{Y-n\theta}{\theta(1-\theta)}.
$$
Its squared <expected value> is $I(\theta)=n/[\theta(1-\theta)]$, using the <binomial distribution> <variance> $n\theta(1-\theta)$. Hence the <Jeffreys prior> is
$$
\boxed{\pi_J(\theta)=\frac1{\pi\sqrt{\theta(1-\theta)}},\quad 0<\theta<1,}
$$
the <Beta distribution> $\operatorname{Beta}(1/2,1/2)$. The factor $\sqrt n$ is independent of $\theta$ and disappears on normalization.
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