Special characterizations of standard discrete models

• This article presents important properties of standard discrete distributions and its conjugate densities. The Bernoulli and Poisson processes are described as genera-tors of such discrete models. A characterization of distributions by mixtures is also introduced. This article adopts a novel singular notation and representation. Singular represen-tations are unusual in statistical texts. Nevertheless, the singular notation makes it simpler to extend and generalize theoretical results and greatly facilitates numerical and computational implementation. Key-Words: • Characterization of discrete distribution; conjugate densities; singular representations