In this article, we present a concise review of developments on discrete multivariate distributions. We first present some basic definitions and notations. Then, we present several important discrete multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
stirling numbers;
regression;
inflated distributions;
truncated forms;
compound distributions;
multinomial;
negative multinomial;
multivariate poisson;
multivariate hypergeometric;
multivariate Polya–Eggenberger;
multivariate discrete exponential;
multivariate power series;
multivariate hermite;
multivariate occupancy;
multivariate weighted;
dirichlet;
multivariate run-related distributions

Discrete Multivariate Distributions

The binomial, negative binomial (and hence geometric) and pisson distributions are characterized, in the context of some well-known bivariate distributions, by the conditional distributionof one random variable (r.v.) given another, and the linear regression of the second, r.v. on the first. The conditional distribution. of the first r.v. given the second is also characterized by each of three sets of conditions.

On characterizing some discrete distributions by linear regression

Multivariate Discrete Distributions

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple ( S , N ) , where N is a count variable and S = X 1 + ⋯ + X N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S | N and N | S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson–Gamma and the Poisson–Binomial conditionals models. Other conditionals models are proposed, including the Poisson–Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.

Joint modelling of the total amount and the number of claims by conditionals

Various techniques for constructing discrete bivariate distributions are scattered in the literature. We review these methods of construction and group them into some loosely defined clusters.

Constructions of Discrete Bivariate Distributions

which may lead to one or more fatalities. It is clear that the number of fatal accidents Y and the number of fatalities Z are both positively correlated with the number of causing accidents X. What is unclear, however, is how sometimes a major accident involving large property damage is not fatal while a minor accident may be fatal. This reflects the uncontrollable nature of those factors which cause traffic fatalities. This suggests that a probability model might be used to express the relationship between the number of accidents and the number of fatal accidents and the relationship between the number of accidents and the number of fatalities. In this investigation, two bivariate probability models will be studied: one for the joint distribution of the number of accidents and the number of fatal accidents; and one for the joint distribution of the number of accidents and the number of fatalities. In both models, the number of accidents X at a certain location during a given time interval is assumed to

M. A. Hamdan

Papers

Some Bivariate Probability Models Applicable to Traffic Accidents and Fatalities