My study group and I are stuck on this Bayes' estimator problem. The question is:
Let X~Pois($\lambda$) Find the Bayes estimator for $\lambda$ with respect to:
(i) The prior distribution: $\lambda$ ~ exp(1)
(ii) The squared error loss function
So for our posterior distribution we ended up with:
$g(\lambda|x_1...x_n)= \dfrac{\frac{\lambda^{\Sigma x_i}}{\Pi x_i!}e^{-\lambda(n+1)}}{\int_0^\infty \frac{\lambda^{\Sigma x_i}}{\Pi x_i!}e^{-\lambda(n+1)}, d\lambda} $
which is a [Gamma distribution][1] with parameters $\Sigma x_i + 1$ and $n+1$.
Then our Bayes' estimator with respect to the Prior and the Squared Error Loss Function is the expected value which we get:
$\frac{\Sigma x_i + 1}{n+1} $
Which we are just not sure if this is right or not.
Could someone please let us know if our thinking is correct on this one?
Thanks!