In this article we propose a weighted and mixed version of binomial and discrete Lindley distributions. The weighted distributions are usually considered for the adjustment of probability of occurrences and removal of biasedness from the data and mixed distributions are usually used in heterogeneous and over-dispersed data sets. The proposed model adjusts the probabilities by removing the bias but contrary to the over-dispersion the model handles the under-dispersion issues. Moreover, various properties like failure rate, moments, factorial moments, recurrence relation between moments, self-decomposability, infinite divisibility and limiting form of the proposed model are also studied. A simulation is conducted and estimation problem is discussed via two estimation methods like maximum likelihood and moment’s estimation. Performance of the proposed model over the binomial, discrete gamma, Poisson-Lindley and discrete Lindley is investigated via some test statistics and two under- dispersed real data sets.
Keywords: Characterization; Poisson distribution; Negative moments; Estimation; Mixture 2000 MSC: 60E05, 62E15