= Solution
{c}
The <prior mean>, <posterior mean>, and <maximum-likelihood estimate> are
$$
\widetilde\theta_0=\frac1{\beta+1},\qquad
\widetilde\theta_1=\frac{S+1}{N+\beta+1},\qquad
\widehat\theta=\frac SN.
$$
For $0<S<N$, the last follows by maximizing $S\log\theta+(N-S)\log(1-\theta)$. At $S=0$ or $S=N$, use the usual <binomial likelihood> convention that admits the endpoints $0$ and $1$ as <maximum-likelihood estimates>; if the parameter domain is strictly $(0,1)$, these are boundary suprema rather than attained maxima. The <posterior mean> remains interior in all cases.
Taking
$$
\boxed{\lambda=\frac{\beta+1}{N+\beta+1}}
$$
gives
$$
\lambda\widetilde\theta_0+(1-\lambda)\widehat\theta
=\frac1{N+\beta+1}+\frac S{N+\beta+1}
=\boxed{\widetilde\theta_1}.
$$
Since $N=nm>0$ and $\beta+1>0$, \b[$0<\lambda<1$]. Thus the <posterior mean> is a <convex combination> of the <prior mean> and observed <binomial proportion>, with larger prior weight when $\beta+1$ is large relative to $N$.
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