Discrete Bimodal Normal Distribution: Theory and Applications to Discrete Failure Lifetime and Count Data

Bimodal discrete distributions provide a flexible statistical framework for modeling data that exhibit two distinct modes, a feature often encountered in practical applications such as reliability analysis, risk assessment, and count data modeling. In this paper, we introduce a novel discrete bimodal distribution derived from the alpha-skew-normal distribution. The proposed model is capable of simultaneously capturing bimodality, skewness, and dispersion patterns without relying on mixture structures, thereby offering a unified alternative to existing approaches. We establish its fundamental distributional properties, including the bimodal failure rate, and provide characterization results that formally connect the continuous alpha-skew-normal distribution with the discrete normal distribution. Parameter estimation is conducted via maximum likelihood methods, and the finite-sample performance of the estimators is investigated through simulation studies. Finally, the practical usefulness of the proposed distribution is demonstrated with applications to real data sets, highlighting its potential as a versatile tool for modeling complex discrete phenomena.