In most cases, the probability mass function {𝑝𝑘}𝑘∈𝕂 is not interesting, since it is difficult to deal with and there is no clear interpretation of the pattern of randomness it describes. The craft of probabilistic modelling (Gani (1986) ^[1]) uses a diversity of criteria to describe and select models, namely, those arising from randomness patterns (such as counts in Bernoulli trials, sampling with or without replacement, and random draws from urns). Another source of the rationale description of count models are characterisation theorems based on structural properties (e.g., a power series distribution with mean = variance, or maximum Shannon entropy with prescribed arithmetic and/or geometric mean). Recurrence relationships (for instance, ) or mathematical properties (for instance, the variance being at most a quadratic function of the expectation) also define interesting families of discrete random variables. On the other hand, asymptotic properties such as arithmetic properties, namely, infinite divisibility, discrete self-decomposability, and stability, serve as guidance in model choice.