= Solution
{c}
For one <Bernoulli random variable>,
$$
\log f(x\mid\theta)=x\log\theta+(1-x)\log(1-\theta),\qquad
\partial_\theta\log f(x\mid\theta)=\frac{x-\theta}{\theta(1-\theta)}.
$$
Its <variance> is $\theta(1-\theta)$, so the expected <Fisher information> and <Jeffreys prior> are
$$
I(\theta)=\frac1{\theta(1-\theta)},\qquad
\pi_J(\theta)\propto\theta^{-1/2}(1-\theta)^{-1/2}.
$$
The normalizer is $B(1/2,1/2)=\pi$. Therefore
$$
\boxed{\pi_J(\theta)=\frac1{\pi\sqrt{\theta(1-\theta)}},\quad0<\theta<1;\qquad
\theta\sim\operatorname{Beta}(1/2,1/2).}
$$
This <Jeffreys prior for a Bernoulli parameter> is proper despite its integrable endpoint singularities. It is the <arcsine distribution>. It replaces the specified $a,b>1$ <prior distribution> in part (a) when the <Jeffreys prior> rule is chosen; it is not one of those restricted priors.
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