= Solution
A <Beta distribution> is a convenient prior for a <probability> $p\in(0,1)$ and is conjugate to the <binomial distribution> of the death count. Match its mean $m=0.05$ and <variance> $v=0.02^2=0.0004$. For $p\sim\operatorname{Beta}(\alpha,\beta)$,
$$
\frac{\alpha}{\alpha+\beta}=m,\qquad
\frac{m(1-m)}{\alpha+\beta+1}=v.
$$
The <moment matching for a beta prior> formula gives
$$
\kappa=\alpha+\beta=\frac{0.05(0.95)}{0.0004}-1=117.75,
\qquad\boxed{\alpha=5.8875,\quad\beta=111.8625.}
$$
Conditional on $p$, use $D\sim\operatorname{Binomial}(90,p)$ and the observed count $D=9$. Multiplying its <likelihood> $p^9(1-p)^{81}$ by the prior density gives the <Bayesian posterior> through <Beta-binomial conjugacy>:
$$
\boxed{p\mid D=9\sim\operatorname{Beta}(14.8875,192.8625).}
$$
Under <squared-error loss>, take its <posterior mean> as the point estimate. A 95% equal-tail <credible interval> is given by its 0.025 and 0.975 quantiles. Numerically,
$$
\boxed{\widehat p_{\mathrm B}=0.07166,\qquad\operatorname{CI}_{0.95}=[0.04076,0.11035].}
$$
Matching two moments does not uniquely determine a prior distribution; the beta family is a justified convenient choice, not a conclusion forced by those moments. Its transfer to this hospital presumes that the historical between-hospital distribution is relevant to the present risk and <case mix>.
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