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

ABSTRACT 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.