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Most people are exposed to sudden death as a result of viral hepatitis infection, especially at an advanced age, so we need to know the continuation of the disease until death in a sample of hospitals to know the differences in the services provided by those hospitals. The research aims to estimate a function for the Rayleigh distribution (RD) using several methods (maximum likelihood function, White, s, weighted least square) and choose the best method for estimation using the mean square error (MSE) and compare the survival rates of patients admitted to (Al-Kadhimiya, Al-Karkh, and Yarmouk) hospitals, The results of the research showed in the experimental aspect that the maximum likelihood method for all sample sizes is the best, followed by the weighted least squares method and finally the White method. In the applied aspect, on a sample consisting of (63) patients (liver cirrhosis + liver cancer) for the first quarter of 2025, it was found that their survival rate until death is (33%) for a time average of (2) months and (15) days.

This research aims to estimate the reliability function of the Rayleigh distribution Under Type-II-Censored Data, which represents. The time of the machine's operation until the failure, and three methods were chosen to estimate the reliability function, which are, the maximum likelihood method, the momentary method, and the least squares method. As the values of the reliability function were calculated for real data related to the machines of theof the machines of Baghdad Pepsi Cola Company, and the comparison was made between the methods of estimating reliability by choosing the largest arithmetic mean for the reliability of the machines, and the greatest possibility appeared the best method, and the researcher concluded that the average times working until holidays is (4) months with a reliability of 87%.

There are many patterns of breast cancer change that make it a global health challenge. The research aims to propose an expanded gamma distribution with parameters and apply it to data for (103) patients with breast cancer. Data normality tests were used, such as the Kolmigrov-Smirnov test, the Anderson-Darling test, and the Chi-square test, to fit the real data. The parameters of the proposed distribution were estimated using the maximum likelihood method. It was found that there is a large difference in the real data with positive Skewness in it. The maximum likelihood estimates reflected the suitability of the data to the proposed distribution, which indicates the accuracy of the obtained results.

The Erlang distribution is a special case of the Gamma distribution with the k shape parameter and the λ rate parameter. In this study, the parameter estimation of the Erlang distribution was carried out using the Maximum Likelihood method. In maximizing the function, an implicit and non-linear form is obtained, then it is solved using the Newton Raphson algorithm. Apart from Newton Raphon, the estimation of parameters was also carried out using the Fisher Scoring algorithm. The Fisher Scoring algorithm is similar to the Newton Raphson algorithm, the difference is that Fisher Scoring uses an matrix information. The result of parameter estimation in Erlang distribution using Newton Raphson algorithm which is applied to outgoing telephone call data that generated by Matlab R2010a software cannot be done simultaneously. Therefore, the parameter assessment is carried out on the k parameter first, then followed by the λ parameter estimation and the parameter and  = 0.6886812 are obtained. Meanwhile, the parameter estimation using the Fisher Scoring algorithm produces an equation that is not different from the Newton Raphson algorithm