In several fields, such as survival analysis, reliability theory, and forensic science, the mean past lifetime (MPL), also known as the expected inactivity time function, plays a vital role. For inference on the MPL function, some procedures have been proposed in the literature, based on a Central Limit Theorem result for the MPL function's estimator. In this thesis, an empirical likelihood (EL) inference procedure of the MPL function is proposed. In addition to that, we obtain the adjusted EL and mean EL confidence interval for the MPL function. The proposed confidence intervals are compared through simulation studies in terms of coverage probability and the average length of the confidence interval. The simulation studies showed that the proposed EL methods have better coverage probability and shorter average lengths than the normal approximation result. Finally, the proposed methods are illustrated by two real data analyses.