The use and application of moment distributions has a great importance in probability, statistics and mathematics in the context of research related to reliability, biomedicine, ecology and several other fields. The concept of moment (weighted) distribution can be traced from the study of the effect of methods of ascertainment upon estimation of frequencies by Fisher (1934). Rao (1965) extended the basic idea of Fisher and introduced moment (weighted) distributions to model such situations. The moment distributions arise in the context of unequal probability sampling. The observations, generated from stochastic process and recorded with some weight function, are needed to model by these distributions. A unifying approach for correction of biases that exist in unequally weighted sample data is provided by Moment (weighted) distribution theory. The hazard rate function has a pivotal role in actuarial sciences, reliability and survival analysis.