# Defining a posterior for poisson distributed data

I am trying to make a formulation of a posterior for a problem with this structure. For a chemistry experiment we have completed two rounds. The first round lasted for 4 months. In this round two cases were observed. The second round lasted 7 months. In this round in the first three months we observed 5 cases and in the last four months we observed 3 cases. The cases observed are distributed with a poisson distribution with parameter $$\lambda$$ with units $$year^{-1}$$. If we denote the cases observed as $$x$$, its distribution is: $$f(x)=\lambda^x e^{-\lambda} (x!)^{-1}$$ And the structure of information would be:

  Run  x  n
1 Run1 2 4/12
2 Run2 5 3/12
3 Run2 3 4/12


The computing of $$n$$ is based in that $$\lambda$$ is in $$year^{-1}$$. Therefore, after setting a prior for $$\lambda$$ such as $$\Gamma(\alpha,\beta)$$ we obtain this posterior, just using information from Run1: $$p(\lambda|x)=\Gamma(\alpha+\sum x_i,\beta+n)$$ If $$\alpha=2$$ and $$\beta=3$$ from prior for $$\lambda$$ and considering Running 1, we would have that posterior is $$\Gamma(2+2,3+\frac{4}{12})$$. Below this scheme considering that $$\lambda$$ is $$year^{-1}$$, is correct to compute $$n$$ considering the number of months each runing lasted? Because this affects on parameters of posterior.