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The Poisson distribution is a discrete probability model, widely used in science and engineering to describe various natural and man-made phenomena. It possesses an important feature, namely being inherently asymmetric, but as its parameter becomes large, the distribution becomes approximately symmetric. To broaden its use, multiple extensions and variations have been developed. Determining whether a data set follows a Poisson distribution involves hypothesis testing at a chosen significance level. When sampling from a Poisson distribution, confidence intervals provide an estimated range instead of a single value. Due to the discrete nature of the Poisson distribution, confidence intervals cannot be derived from a simple formula, and are therefore computed using specialized algorithms. In this paper, three alternatives are given and discussed.

Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), the etiological agent of the Coronavirus Disease 2019 (COVID-19) disease, has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used—appropriately and less appropriately—to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, often dominated by a small number of individuals, and heavily influenced by superspreading events (SSEs). The distinct transmission features of SARS-CoV-2, e.g., high stochasticity under low prevalence (as compared to other pathogens, such as influenza), and the central role played by SSEs on transmission dynamics cannot be overlooked. Many explosive SSEs have occurred in indoor settings, stoking the pandemic and shaping its spread, such as long-term care facilities, prisons, meat-packing plants, produce processing facilities, fish factories, cruise ships, family gatherings, parties, and nightclubs. These SSEs demonstrate the urgent need to understand routes of transmission, while posing an opportunity to effectively contain outbreaks with targeted interventions to eliminate SSEs. Here, we describe the different types of SSEs, how they influence transmission, empirical evidence for their role in the COVID-19 pandemic, and give recommendations for control of SARS-CoV-2.

Poisson regression is a popular tool for modeling count data and is applied in a vast array of applications from the social to the physical sciences and beyond. Real data, however, are often over- or under-dispersed and, thus, not conducive to Poisson regression. We propose a regression model based on the Conway—Maxwell-Poisson (COM-Poisson) distribution to address this problem. The COM-Poisson regression generalizes the well-known Poisson and logistic regression models, and is suitable for fitting count data with a wide range of dispersion levels. With a GLM approach that takes advantage of exponential family properties, we discuss model estimation, inference, diagnostics, and interpretation, and present a test for determining the need for a COM-Poisson regression over a standard Poisson regression. We compare the COM-Poisson to several alternatives and illustrate its advantages and usefulness using three data sets with varying dispersion.

Studying the positive definiteness of the covariance matrix of discrete samples helps to determine whether the dimensionality of the samples can be reduced, which is beneficial for optimizing the number of samples and designing optimal plans for sampling surveys. This paper aims to provide a method to determine the variable numbers of the sample subjecting to Poisson distribution. Methods . It is based on the theory of I -linear combination and its properties which are the author’s previous studying results. Results . study shows the covariance matrix of multi-Poisson distribution is positively defined and the probability of the sample covariance matrix of multi-poisson distribution is about 1 when the sample capacity is very large. Conclusion . The dimension size of the sample data matrix of multi-poisson distribution can be reduced when the sample capacity n is no more than the dimension size p .

In this note, the dispersion of -Poisson distribution is studied. It is specifically shown that this distribution is over-dispersed when and under-dispersed when .

This article aims to gain implicit relations among different subclasses of harmonic family of analytic functions containing with some description for Poisson distribution series to be in the new subclasses of harmonic convex functions and harmonic starlike, ( α ) and U W * ( α ) respectively, in the plane.

In this article, new properties of the Poisson distribution of order k with parameter λ are found. Based on them, the modes of the Poisson distributions of order k=3 and 4 are derived for λ in (0,1). They are 0, 3, 5, and 0, 4, 7, 8, respectively, for λ in specified subintervals of (0, 1). In addition, using Mathematica, computational results for the modes of the Poisson distributions of order k=2,3, and 4 are presented for λ in specified subintervals of (0,2).

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients a2 and a3, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results.

Methodological developments in different sectors like health, biomedical and biological areas are the recent burning issue in the statistical literature. The approach of implementing declining hazard function which is obtained by compounding truncated Poisson distribution and a lifetime distribution is a special concern in a few studies. In this paper we proposed a newly introduced distribution called inverse Lomax-Uniform Poisson distribution mostly applied in the health, biomedical, biological, and related sectors. Some basic statistical properties of the distribution are discussed. The capability of the model is checked by comparing it with six potential models by using a practical real data set. Based on the comparison techniques, the newly proposed model out performs all its counterparts. The simulation study is also conducted. Furthermore, the joint modelling of repeatedly measured and time-to-vent processes is discussed in detail with the real data set in the health sector.

By using the generalization of the neutrosophic q-Poisson distribution series, we introduce a new subclass of analytic and bi-univalent functions defined in the open unit disk. We then apply the q-Gegenbauer polynomials to investigate the estimates for the Taylor coefficients and Fekete–Szegö type inequalities of the functions belonging to this new subclass. In addition, we consider several corollaries and the consequences of the results presented in this paper. The neutrosophic q-Poisson distribution is expected to be significant in a number of areas of mathematics, science, and technology.