A park of area $S=10 000 km^2$ was surveyed for bears, and out of $n$ disjoint regions of equal area $s=1km^2$, there were $n_k$ regions with $k=0,1,....,N$ bears. On each of these regions, the amount of bears can be modeled by a Poisson distribution with an unknown but constant parameter $\mu$. I want to find an estimate for $\mu$ using maximum likelihood, but I'm not sure how my function should look like. My idea is this:
$$\prod_{k=1}^{N}\left( P(N_k=k) \right) ^{n_k}$$
where the probability is Poisson, and $N_k$ is the amount of bears in area $k$, but I have no clue if its correct or not. I think I'm mixing up symbols. I always have trouble constructing the likelihood function in problems such as this.
Also, given some sample of $n_k$'s, I then have to check whether the data is consistent with the theoretical model. I'm guessing I have to do some hypothesis testing, whether the sigmas and mu's are equal, is that right?