An observable r.v. $X$ is Poisson with unknown mean $\theta$. Under squared error loss: Find the Bayes Rule for estimating $\theta$, when $\theta$ has a prior gamma density $\pi$ with known index $\alpha$ and parameter $\lambda$. Find the risk function of the Bayes risk and the Bayes risk of the prior $\pi$.
The problem is that I try to find the posterior distribuion but I cannot recognise it as any known function. I guess it is a Gamma distribution but I don't know how to identify its parameters. Can anybody help me with that, please?
Thank you in advance.