In this paper, we propose and study a new probability mass function by creating a natural discrete analog to the continuous Lindley distribution as a mixture of geometric and negative binomial distributions. The new distribution has many interesting properties that make it superior to many other discrete distributions, particularly in analyzing over-dispersed count data. Several statistical properties of the introduced distribution have been established including moments and moment generating function, residual moments, characterization, entropy, estimation of the parameter by the maximum likelihood method. A bias reduction method is applied to the derived estimator; its existence and uniqueness are discussed. Applications of the goodness of fit of the proposed distribution have been examined and compared with other discrete distributions using three real data sets from biological sciences.

中文翻译：

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具有可靠性应用的连续 Lindley 分布的新离散模拟

在本文中，我们通过将连续林德利分布作为几何和负二项式分布的混合创建自然离散模拟，提出并研究了一种新的概率质量函数。新分布具有许多有趣的特性，使其优于许多其他离散分布，尤其是在分析过度分散的计数数据方面。已经建立了引入分布的几个统计特性，包括矩和矩生成函数、残差矩、表征、熵、通过最大似然法估计参数。将偏差减少方法应用于导出的估计量；讨论了它的存在性和唯一性。