I have been looking everywhere for the solution to this problem so here's to hoping some of you can help me.

The number of cases X of a rare disease is assumed to follow a Poisson distribution, X ~ Poi(lambda)

We want to analyse the prevalence of the disease closer. To estimate the probability p of having a child with the disease we observe the number of children born n_i and the number of cases of the disease X_i for every year i where i=1,2,...,m. We assume that the probability is constant in these m years. Also, n_i is large and p is small, so we can assume that X_i ~Poi(n_i * p), i=1,2,...,m.

So, how do we find the MLE of p, what is it, it's expectation and its variance?

Thanks in advance for any help!