A Generalization of the Poisson Distribution

Abstract

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Spatial galaxy distribution function for a three-component system

“…Further, it may be noted that for m 1 = m 2 = m 3 (i.e. all components having the same mass), (25) reduces to the well known "Generalized Poisson Distribution" (Consul and Jain 1973):…”

Another generalization of the geometric distribution

“…It is also possible to perform a direct search of the maximum likelihood function to obtain the maximum likelihood estimator; this can be done using software such as Mathematica or Matlab. The data sets presented in Table 2 (accidents to 647 women working on H. E. Shells for 5 weeks, discussed in Consul and Jain (1973)) and Table 3 (number of carious teeth among the four deciduous molars in a sample of 100 children aged 10 and 11 years (Krishna and Singh 2009) are overdispersed since the sample variances are greater than the respective sample means. Because of the overdispersion phenomena, distributions with this characteristic were also used to compare to the GG distribution.…”

“…They are the negative binomial (NB) with parameters r > 0, 0 < 1 < p; the generalized Poisson distribution (GP) with parameters λ 1 > 0, |λ 2 | < 1; and the Poisson-inverse Gaussian distribution (PIG) (see Willmot 1987) with parameters μ > 0, φ > 0. Furthermore, the data sets presented in Table 4 (results of ten shots fired from a rifle at each of 100 targets in Consul and Jain 1973) and Table 5 (fish catch data in Kemp 1992) are infradispersed since the sample variances are lower than the respective sample means. In this example, only the generalized Poisson distribution (which can be overdispersed or infradispersed) was considered to compare to the results obtained using the GG distribution.…”

Asymptotic behaviour of Jain operators

“…Note that this is a probability distribution which is called generalized Poisson distribution. It has been considered by Consul and Jain [7,8] and several other authors. In 1985, Umar and Razi [22] defined a Kantorovich-type integral modification of the operators (1).…”