# One formula for the probabilities of the poisson, binomial, and negative binomial distribution

## One formula for the probabilities of the Poisson, Binomial, and Negative Binomial distribution

### Michael Fackler

Abstract:

This paper gives a formula representing all discrete loss distributions of the Panjer class (Poisson, Binomial, and Negative Binomial) in one. Further it provides an overview of the many Negative Binomial variants used by actuaries.

Keywords:

Panjer class, (a,b,0) class, discrete loss distribution, Negative Binomial

Resumen:

En este artículo se presenta una fórmula conjunta para las probabilidades de las distribuciones de la clase Panjer (Poisson, binomial, binomial negativa). Además se da una mirada general a las variantes de la distribución binomial negativa utilizadas por los actuarios. Palabras clave. Clase de Panjer, clase (a,b,0), distribución del número de siniestros, distribución binomial negativa 1. Introduction The three well-known discrete loss distributions Poisson, Binomial, and Negative Binomial are closely related. First of all, they form the Panjer (a,b,0) class (see Panjer and Willmot, 1992, section 7.2; Klugman et al., 2004, appendix B.2). Secondly, the Poisson distribution is a limiting case of the two other distributions, which finally have their origin in the modelling of Bernoulli trials. The traditional representations of the probability (mass) functions of the three distributions look quite different, though. In this paper we add to the unified view of these distributions by presenting a common formula for the probabilities, being both instructive and convenient for practical implementation.

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