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.

Many thanks for your help.


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