A Note on the (Weighted) Bivariate Poisson Distribution

Abstract

In the recent statistical literature, the univariate Poisson distribution has been generalized by many authors, among them: the univariate weighted Poisson distribution [13], the generalized univariate Poisson distribution [7], the bivariate Poisson distribution according to Holgate [11], the bivariate Poisson distribution according to Lakshminarayana, Pandit and Srinivasa Rao [15], the bivariate Poisson distribution according to Berkhout and Plug [4], the bivariate weighted Poisson distribution according to Elion et al. [8] and the generalized bivariate Poisson distribution according to Famoye [9]. In this paper, We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distributions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout and Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N.

Comparative Analysis of Ultrasonic and Traditional Gas-Leak Detection Systems in the Process Industries: A Monte Carlo Approach

“…Sensor detection probabilities 1, 2, and 3 represent the detection probabilities of the concentration-measurement, ultrasonic, and combined gas-leak detectors, respectively. The Poisson distribution, a probability distribution based on the likelihood of certain events occurring within a specific time period [53][54][55][56], was applied to the combined detector model in our study. Here, 'events' refer to instances where smoke is undetected.…”

The Bivariate Extended Poisson Distribution of Type 1

“…Several authors have studied bivariate Poisson laws, in particular, Berkhout and Plug [2] and Lakshminarayna et al [7]. Then, [3] has highlighted the weighted bivariate Poisson law having as a basic law, the bivariate Poisson law according to Berkhout and Plug [2]; a law that allows to generate all the bivariate Poisson laws. The bivariate Poisson distribution according to Berkhout and Plug [2] is rightly considered the standard distribution in N 2 as is the Poisson distribution in N. In this paper, we will construct the bivariate extended Poisson distribution which generalizes the univariate extended Poisson distribution.…”