The Poisson distribution has a well-known axiomatization:
Events are occurring independently.
The average rate (number of occurrences in a unit period) is constant.
Two events cannot occur at exactly the same instant.
The probability of more than one occurrence in a sufficiently small period is negligible.
My question is, what is the relevant literature,i.e. a good book, that concretely axiomatizes dozens of known p.d.f. each by several different set of axioms at best ?