Solution
= Solution
The posterior is $\operatorname{Gamma}(α+X,β+1)$, so under quadratic loss the Bayes estimator is $δ(X)=(α+X)/(β+1)$. At fixed $θ$ its risk is
$$
\boxed{R(θ,δ)={θ+(α-βθ)^2\over(β+1)^2}.}
$$
= Solution
The posterior is $\operatorname{Gamma}(α+X,β+1)$, so under quadratic loss the Bayes estimator is $δ(X)=(α+X)/(β+1)$. At fixed $θ$ its risk is
$$
\boxed{R(θ,δ)={θ+(α-βθ)^2\over(β+1)^2}.}
$$