Covariate-dependent stress–strength reliability under the shape-extended continuous Bernoulli distribution

In this article, we extend the continuous Bernoulli to a shape-flexible distribution on the unit interval (0,1). Unlike the well-known beta distribution, it admits a closed-form cumulative distribution function (CDF). The extended model preserves the analytical tractability of the original continuous Bernoulli distribution while allowing greater flexibility. We derive explicit expressions for the moments and, most notably, for the stress–strength reliability function in closed form. Maximum likelihood estimation is implemented via efficient fixed-point and scoring-based routines. We then develop a logistic link regression that treats stress–strength reliability as a covariate-dependent quality index for bounded outcomes, yielding interpretable reliability curves. An illustrative application demonstrates that the model captures complex reliability patterns and offers practical utility for reliability and quality engineering.