Solution
= Solution
Put $c=(β+1)/(β+2)$. The posterior mean of $e^{-θ}$ is $δ(X)=c^{α+X}$. Using the Poisson probability-generating function, its risk is
$$
\boxed{c^{2α}e^{θ(c^2-1)}-2c^αe^{θ(c-2)}+e^{-2θ}.}
$$
= Solution
Put $c=(β+1)/(β+2)$. The posterior mean of $e^{-θ}$ is $δ(X)=c^{α+X}$. Using the Poisson probability-generating function, its risk is
$$
\boxed{c^{2α}e^{θ(c^2-1)}-2c^αe^{θ(c-2)}+e^{-2θ}.}
$$