Abstract
Differential equations are used in modelling many disciplines, in engineering, chemistry, physics, biology, economics, and other fields of sciences, hence can be used to understand and to determine the underlying probabilistic behavior of phenomena through their probability distributions. This paper came to use a simple form of differential equations, namely, the linear form, to determine the probabilistic distributions of some of the most important and popular sub class of discrete distributions used in real-life, the Poisson, the binomial, the negative binomial, and the logarithmic series distributions. A class of finite number of inflated points power series distributions, that contains the Poisson, the binomial, the negative binomial, and the logarithmic series distributions as some of its members, was defined and some of its characteristics properties, along with characterization of the 3-points inflated of these four distributions, through a linear differential equation for their probability generating functions were given. Further, some previous known results were shown to be special cases of our results.
Highlights
A discrete random variable Y is said to have a power series distribution (PSD), abbreviate that by writing Y ∼ PSD(θ, g(θ)), if its probability mass function (PMF) is given by; P(Y = y) = ayθy g(θ), y ∈ T (1)where g(θ) = ∑y∈T ay θy is a power series for θεΩ = {θ; 0 < θ < ω}, ω is the radius of convergence of g(θ), and ay ≥ 0 for all y ∈ T ⊆ I = {0,1,2, ... }
If we look at the above frequencies of the number of intervals, we notice that the number of fetal movement with zeros is inflated and the ones, and even the twos may be inflated
After defining a class of finite number of inflated points power series distributions in Section 2 along with some of its characteristics properties, we considered characterization of the 3-points inflated of the four well-known discrete distributions; namely, the Poisson, binomial, negative binomial, and the logarithmic series distributions, through a linear differential equation for their probability generating functions in Sections 3, 4, 4 and 6, respectively
Summary
Phang and Loh (2013), reviewed some related literature zero inflated models for over dispersed count data and provide a variety of examples from different disciplines in the applications of zero inflated models, as well as discussed using different model selection methods in model comparison. Zamri and Zamzuri (2017), review related literature to the zero inflated models, provide a recent development and summary on models for count data with extra zeros. After defining a class of finite number of inflated points power series distributions in Section 2 along with some of its characteristics properties, we considered characterization of the 3-points inflated of the four well-known discrete distributions; namely, the Poisson, binomial, negative binomial, and the logarithmic series distributions, through a linear differential equation for their probability generating functions in Sections 3, 4, 4 and 6, respectively
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