## 1 Introduction  
[PERSON] and [PERSON] [1] introduced the moment-based exponential distribution. [PERSON] et al. [2] introduced and studied properties of the exponentiated moment-based exponential distribution. The cumulative distribution function of the exponentiated moment-based exponential distribution is as follows:  
\[F_{X}(x;\alpha,\beta)=\left(1-\left(1+\frac{x}{\beta}\right)\!e^{-\frac{x}{ \beta}}\right)^{\alpha},\text{ for }x>0, \tag{1}\]  
and its probability density function is as follows:  
\[f_{X}(x;\alpha,\beta)=\frac{\alpha xe^{-\frac{x}{\beta}}}{\beta^{2}}\left(1- \left(1+\frac{x}{\beta}\right)\!e^{-\frac{x}{\beta}}\right)^{\alpha-1},\text{ for }x>0, \tag{2}\]  
where \(\alpha\) is the shape parameter and \(\beta\) the scale parameter. The moment distributions have a wide range of applications in different fields of study. [PERSON] and [PERSON] [3] and [PERSON] [4] examined the relationship between particle diameter and its frequency using moment distributions. [PERSON] [5] used canonical distributions to characterize the relationship of land birds in England and Wales, breeding birds in Finland, mammals in the East Indies, and amphibian and reptiles in the West Indies. [PERSON] [6] introduced the use of weighted distributions to represent length-biased sampling in the case of cell kinetics and the early detection of disease. In addition,Zelen [7]) used length-biased sampling in the context of the disease screening and scheduling of patients' examinations. [PERSON] [8] introduced low-density traffic streams in the context of studying various aspects of the traffic streams. [PERSON] [9] studied the role of statistical distribution in different cases related to forestry and forest products. [PERSON] et al. [10] used weighted distributions in the context of modelling populations of fish stocks. Let \(G\) be the cumulative distribution function of \(Y\) and \(F\) be the cumulative distribution function of \(X_{1},X_{2},\ldots X_{k}\). [PERSON] and [PERSON] [11] developed multicomponent reliability for a stress-strength model as follows:  
\[ \begin{array}{l}R_{s,k}=P(\text{at least s variables among }X_{1},X_{2},\ldots X_{k}\text{ exceed }Y)\\ \qquad\qquad\qquad=\sum\limits_{i=s}^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\int\limits_{-\infty}^{\infty}{(1-G(y))^{i}(G(y))^{k-i}dF( y)},\end{array}  \tag{3}\]  
where \(X_{1}\), \(X_{2}\),..., \(X_{k}\) are independent and identically distributed random variables with cumulative distribution function \(F\). This system has a common random stress \(Y\). The probability, \(R_{s,k}\), in Equation (3) is called reliability in a multicomponent stress-strength model. [PERSON] and [PERSON] [12] estimated the probability of \(Y<X\). [PERSON] [13] extended their results to the normal distribution; [PERSON] and [PERSON] [14] to the Burr distribution; [PERSON] [15] to the Weibull; [PERSON] and [PERSON] [16] to some distributions useful in lifetesting (a process done under controlled conditions to determine how and when a product will fail); [PERSON] and [PERSON] [17] developed inference for reliability and stress-strength for a scaled Burr type \(X\) distribution;and [PERSON] and [PERSON] [18] compared estimators of \(P(Y<X)\) for a Burr type \(X\) distribution. Their work was extended by [PERSON] and [PERSON] [19] for the exponential distribution, by [PERSON] and [PERSON] [20] for the Weibull distribution, by [PERSON] et al. [21] for the three-parameter exponential distribution, and by [PERSON] and [PERSON] [22] for the three-parameter Weibull distribution. [PERSON] and [PERSON] [11] expressed reliability in a multicomponent stress-strength model, as did [PERSON] and [PERSON] [23] with the Burr distribution. [PERSON] and [PERSON] [24] estimated reliability in the multicomponent stress-strength model for a log-logistic distribution, [PERSON] [25] for the exponential distribution, [PERSON] et al. [26] for an inverse Rayleigh distribution, and [PERSON] et al. [27] for the Burr XII distribution.