Christian Lorentzen talks about Tweedie distributions:
Tweedie distributions and Generalised Linear Models (GLM) have an intertwined relationship. While GLMs are, in my view, one of the best reference models for estimating expectations, Tweedie distributions lie at the heart of expectation estimation. In fact, basically all applied GLMs in practice use Tweedie distributions with three notable exceptions: the binomial, the multinomial and the negative binomial distribution.
Read on for a bit more about its history and how it ties in with several other distributions.