A thesis submitted to the council of the college of Administration &Economics \ University of Karbala as partial fulfillment of the requirements for the degree of Master of Statistics Science

By
Sanaa Ali Mohammed Alabodi
Supervised by
Ass.Prof.Dr. Mahdi Wahhab Noma Nasrallah

Abstract
The theoretical part of this thesis involves the derivation of the mathematical formula of the cascade system for more than one component that has the strength (x) and getting independent stress ( with an attenuation factor (k) which works to recertification the path of this system and make it works in right way. Also, this work reached the reliability function of the Cascade system  depending on the derivative mathematical formula of the inverse Lindley distribution.
After that,  reliability of the Cascade system was then estimated for an inverse Lindley distribution using four methods (Maximum Likelihood Estimation (MLE), Median Method (med), Herd-Johnson (HJ) and Kaplan-Meier (KM).
For comparing the estimations of the reliability of the Cascade system , the Monte-Carlo simulation technique was used by using the statistical measure (MSE) to determine a suitable estimator.
Moreover, it was concluded that the reliability of  is increasing by increase the value of the strength parameter, while is decreasing by increase the value of the stress parameter. The results of the simulations showed that the best estimate was at the size of the sample n = 150 using MLE measuring, even with small percentage of difference with (Med) method.
For applied side of this thesis, the researcher used what was reached in the experimental part for estimate the reliability function of the Cascade system in the State Company for Fertilizer Industry in Basra.