A new one-parameter discrete distribution, namely discrete Burr-Hatke distribution is introduced and its mathematical properties are studied comprehensively. The main properties of the discrete Burr-Hatke distribution such as mean, variance, skewness and kurtosis measures are obtained in explicit forms. Several parameter estimation methods are used to obtain unknown model parameters and these estimation methods are compared via simulation study. The discrete Burr-Hatke distribution is over-dispersed since its variance is greater than its mean. This property of the proposed distribution opens a new opportunity to model over-dispersed data sets. To show the importance of the proposed distribution against the existing discrete probability distributions, three data sets in different fields are analyzed. Additionally, count regression model based on the discrete Burr-Hatke distribution is introduced with its residual analysis.

In this article, the stochastic Hirota–Maccari system forced in the Itô sense by multiplicative noise is considered. We just use the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method to get new stochastic solutions which are hyperbolic, trigonometric, and rational. The major benefit of these three approaches is that they can be used to solve similar models. Furthermore, we plot 3D surfaces of analytical solutions obtained in this article by using MATLAB to illustrate the effect of multiplicative noise on the exact solution of the Hirota–Maccari method.

This paper introduces a new two-parameter exponentiated discrete Lindley distribution. A wide range of its structural properties are investigated. This includes the shape of the probability mass function, hazard rate function, moments, skewness, kurtosis, stress–strength reliability, mean residual lifetime, mean past lifetime, order statistics and L-moment statistics. The hazard rate function can be increasing, decreasing, decreasing–increasing–decreasing, increasing–decreasing–increasing, unimodal, bathtub, and J-shaped depending on its parameters values. Two methods are used herein to estimate the model parameters, namely, the maximum likelihood, and the proportion. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood and proportion estimators. The flexibility of the proposed model is explained by using four distinctive data sets. It can serve as an alternative model to other lifetime distributions in the existing statistical literature for modeling positive real data in many areas.