International Journal of Probability and Statistics 2017, 6(1): 1-10 DOI: 10.5923/j.ijps.20170601.01 The Discrete Poisson-Akash Distribution Rama Shanker Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea Abstract A Poisson-Akash distribution has been obtained by compounding Poisson distribution with Akash distribution introduced by Shanker (2015). A general expression for the r th factorial moment has been derived and hence the first four moments about origin and the moments about mean has been obtained. The expressions for its coefficient of variation, skewness and kurtosis have been obtained. Its statistical properties including generating function, increasing hazard rate and unimodality and over-dispersion have been discussed. The maximum likelihood estimation and the method of moments for estimating its parameter have been discussed. The goodness of fit of the proposed distribution using maximum likelihood estimation has been given for some count data-sets and the fit is compared with that obtained by other distributions. Keywords Akash distribution, Compounding, Moments, Skewness, Kurtosis, Estimation of parameter, Statistical properties, Goodness of fit 1. Introduction A lifetime distribution named, “Akash distribution” having probability density function ( ) ( ) 3 2 2 , 1 ; 0, 0 2 x f x x e x θ θ θ θ θ − = + > > + (1.1) has been proposed by Shanker (2015) for modeling lifetime data from biomedical science and engineering and shown that the proposed distribution is a two-component mixture of exponential distribution having scale parameter θ and a gamma distribution having shape parameter 3 and scale parameter θ with their mixing proportions 2 2 2 θ θ + and 2 2 2 θ + respectively. Various mathematical and statistical properties of Akash distribution including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure and stress-strength reliability have been discussed by Shanker (2015). It has been shown by Shanker (2015) that Akash distribution provides much closer fit than Lindley and exponential distributions for modeling lifetime data from medical science and engineering. Further, Shanker et al (2016) have done a detailed comparative study on Akash, Lindley and exponential distributions for modeling different types of * Corresponding author: shankerrama2009@gmail.com (Rama Shanker) Published online at http://journal.sapub.org/ijps Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved lifetime data from engineering and medical science and concluded that Akash distribution has some advantage over Lindley and exponential distributions, Lindley distribution has some advantage over Akash and exponential distributions and exponential distribution has some advantage over Akash and Lindley distributions due to their over-dispersion, equi-dispersion, and under-dispersion for various values of their parameters. Recently, Shanker and Shukla (2016) has introduced a two-parameter weighted Akash distribution (WAD) andstudied its various mathematical and statistical properties, estimation of parameter and application for modeling lifetime data. Shanker (2016) has also proposed a two-parameter quasi Akash distribution and studied its statistical and mathematical properties, estimation of parameters using maximum likelihood estimation and method of moments along with its application for modeling lifetime data from engineering and biomedical sciences. In the present paper, a Poisson mixture of Akash distribution introduced by Shanker (2015) named, “Poisson-Akash distribution (PAD) has been proposed. Its various mathematical and statistical properties including its shape, moments, coefficient of variation, skewness, and kurtosis have been discussed. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments. The goodness of fit of PAD along with Poisson distribution and Poisson-Lindley distribution (PLD), a Poisson mixture of Lindley (1958) distribution and introduced by Sankaran (1970), has been given with some count data-sets. 2. Poisson-Akash Distribution Suppose the parameter λ of Poisson distribution follows Akash distribution (1.1). Then the Poisson mixture of Akash