The study seeks to use the (T-X Family) method in building a new proposed probabilistic model known as the Maxwell-Rayleigh Distribution with two parameters (𝜃, 𝜆), as some of its properties were studied, its parameters were estimated, and the reliability function was calculated using four estimation methods (Maximum Likelihood Method “MLE”, moment method “MOM”, weighted least squares method “WLS” and Percentiles Estimators method “PC”), and for the purpose of comparison between estimation methods for parameters and reliability function, the Monte-Carlo simulation method was employed. Using the program (Wolfram Mathematica 12.2) to conduct several experiments with different sample sizes (small “25,50”, medium “100” and large “150” and through the use of the statistical scale Mean Error Squares (MSE) with respect to estimating parameters and mean integral error squares (IMSE) As for the estimators of the reliability function, the results showed the preference of the weighted least squares method in estimating parameters and calculating the estimations of the reliability function for the proposed distribution at medium and small sample sizes, and the preference of the Maximum Likelihood Method at large sample sizes.
The proposed distribution was applied to real data with (91) observations representing the working times of the spinning machines until failure, and through goodness of fit tests, it was proven to be superior in representing and describing these data compared to the Maxwell and Riley distributions, and the reliability function of the real data was estimated using the best method that was reached. On the experimental side for the average samples (weighted least squares
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method), it was found that the average operating times until failure of the machines amounted to (6.046484) months, and that the average values of the estimated reliability function amounted to (0.506831), that is, it is possible to rely on these machines at a rate of (50%) in about six months.