Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 28J / b / Solution

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A gamma prior with shape $\alpha$ and rate $\beta$ combined with $n$ independent observations from an <exponential distribution> of rate $\theta$ gives the posterior
$$
\theta\mid X_1,\ldots,X_n
\sim\operatorname{Gamma}\left(\alpha+n,\beta+\sum_{i=1}^nX_i\right).
$$
Under <squared-error loss>, the <Bayes estimator> is its posterior mean $(\alpha+n)/(\beta+\sum_iX_i)$.

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 Incoming links: Gamma-exponential conjugacy