A Study of Two Parameter Based an Exponential Probability Distribution: Properties and Applications

ABSTRACT Defined on negative as well as positive domain, we introduce a novel two‐parameter survival distribution to offer enhanced flexibility and applicability in statistical modeling in the article understudy. This distribution is characterized by highly adaptable probability density with hazard rate function. Specifically, the probability density function can exhibit unimodal behavior with an inverted shape and a light left tail, or it may be monotonically decreasing with a heavy right tail. The hazard rate function demonstrates a wide range of failure pattern, including increasing, decreasing and an unconventional decreasing‐increasing shapes. To evaluate the performance and suitability of the proposed model, several statistical criteria and goodness‐of‐fit measures are employed. Graphical tools such as Probability‐Probability and Quantile‐Quantile plots are also used for model validation. The theoretical development includes an in‐depth examination of the quantile function and a comprehensive analysis of the moments, including mean, variance, standard deviation, covariance, skewness, and kurtosis. Further, the study explores several key statistical properties of the Two Parameter Based Exponential distribution, including order statistics, with particular focus on extreme values, reversed order statistics, upper record statistics residual lifetime function and reversed residual life function. Parameter estimation is conducted using the method of maximum likelihood. A Monte Carlo simulation study is performed to assess the efficiency and accuracy of the estimation procedure. Moreover, two real‐life datasets are analyzed to compare the proposed model against existing distributions. These applications highlight the flexibility and practical relevance of the TPBE distribution in survival analysis and reliability engineering.