The weighted Lindley exponential distribution and its related properties

Both the exponential and Lindley distributions can be used to model the lifetime of a system or process, as well as the distribution of waiting times. In this study, we introduce the $WLE(\theta ,\lambda ,\alpha )$ notation for the weighted Lindley exponential distribution. Using two distinct asymmetrical distributions, the skewness mechanism of Azzalini was implemented in this distribution. In other words, we multiplied the density function of the Lindley distribution by the distribution function of the exponential distribution after adding the skewness parameter $\alpha >0$. This $WLE$ distribution contains the Lindley ^[1], the two parameters weighed Lindley ^[2] and the new weighted Lindley ^[3] distributions as special cases. We investigated the proposed model's mathematical properties. In addition to studying the central moments, we also investigate maximum likelihood estimators. To demonstrate the superiority of our model, we employ the MLE method to fit the weighted Lindley exponential model to the actual data set.