<p>In this paper, we studied some properties of the extended Lindley distribution. In particular, we established a new formula of its non-centered moments. The proof of this formula was constructed using several technical propositions. It was also numerically validated through several simulations and experiments. In addition, we expressed these moments as an expectation of a transformed Gamma random variable. The latter expression provides an alternative way to numerically compute these moments. Moreover, we detailed the steps of deriving the maximum likelihood estimators of the model parameters using both complete and censored data. To evaluate the performance of the proposed algorithm, a simulation study was conducted. Furthermore, the extended Lindley distribution was used in two real-world applications: a non-censored data set and a right censored one both in the context of system reliability. The obtained models were compared to other competitive models in both cases.</p>