I think the following lectures notes, which I used at SPARC, would be helpful for these sorts of analyses. In particular, the opening section on moment generating functions and the section on the Poisson distribution (although the examples in that section got nuked because the dataset I used was proprietary to Dropbox and I haven't yet asked for permission to use it beyond SPARC). Apologies for any roughness (if you find typos please let me know).