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‎The equilibrium distributions have many applications in reliability theory, stochastic orderings and random processes. ‎The purpose of this paper is to introduce the equilibrium distributions and presents some results related to this issue. Some results are based on order statistics. ‎In this paper, ‎the generalized Pareto distributions are also analyzed and some basic relationships between the equilibrium distributions are presented.

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In reliability theory, some measures are introduced , called importance measures, to evaluate the relative importance
of individual components or groups of components in a system. Importance measures are quantitive criteria
that ranke the components according to their importance. In the literature, different importance measures are presented
based on different scenarios. These measures can be determined based on the system structure, reliability of
the components and/or component liftime distributions. The purpose of this paper is the study different importance
measures of the components of a system in reliability theory.

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‎In this paper‎, ‎the concept of joint reliability importance (JRI) of two or groups of components in a coherent system with independent components have been studied‎. ‎The JRI is defined as the rate at which the system reliability improves as the reliabilities‎ ‎of the two or groups of components improve‎.

‎Generally‎, ‎the sign and the value of the JRI represent the type and the‎ ‎degree of interactions between components with respect to systems reliability‎.

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‎In this paper‎, ‎a new distribution is introduced‎, ‎which is a generalization of a well-known distribution‎. ‎This distribution is flexible and applies to income data modeling‎. ‎We first provide some of the mathematical and distributional properties of this new model and then‎, ‎to demonstrate the flexibility the new distribution‎, ‎we will present the applications of this distribution with real data‎. ‎Data fitting results confirm the appropriateness of this new model for the real data set‎.

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In this paper, ‎reliability in multi-component stress-strength models‎, ‎when the stress and strength variables are inverse Rayleigh distributions with different parameters of alpha and beta. ‎Estimates of the maximum likelihood‎, ‎Bayesian and empirical Bayesian are estimated‎. ‎Then‎, ‎with the help of Monte Carlo simulation and two real data sets‎, ‎these estimation methods are compared‎.

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Consider a coherent system consisting of independent or dependent components and assume that the components are randomly chosen from two different batches, where the components lifetimes of the first batch are larger than those of the second in some stochastic order sense. In this paper, using different stochastic orders, we compare the reliability of such systems and show that the reliability of the systems increases, as the random number of components chosen from the first batch increases in different stochastics orders.  We use copula function to describe dependence structure between component lifetimes.

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In this paper, the difference between classical regression and fuzzy regression is discussed. In fuzzy regression, nonphase and fuzzy data can be used for modeling. While in classical regression only non-fuzzy data is used.
The purpose of the study is to investigate the possibility of regression method, least squares regression based on regression and linear least squares linear regression method based on fuzzy weight calculation for non-fuzzy input and fuzzy output using symmetric triangular fuzzy numbers. Further reliability, confidence intervals and fitness fit criterion is presented for choosing the optimal model.
Finally, by providing examples of the behavior of the proposed methods, the optimality of the regression hybrid model is shown by the least linear fuzzy squares.

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The Kumaraswamy distribution is a two-parameter distribution on the interval (0,1) that is very similar to beta distribution. This distribution is applicable to many natural phenomena whose outcomes have lower and upper bounds, such as the proportion of people from society who consume certain products in a given interval. 

In this paper, we introduce the family of Kumaraswamy-G distribution, and we detect its hazard rate function, reversible hazard rate function, mean residual life function, mean past life function, and the behavior of each of them. Also, we investigate the stochastic order concept of the family of Kumaraswamy-G distribution. Finally, in the form of a practical example, we analyze the suitability of the Kumaraswamy distribution for real data.

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The purpose of the present study was to examine the factor validity and reliability of the questionnaire of student perception of assessment and evaluation task [19]. In this regard, the questionnaire of students’ perception of the assessment and evaluation task was performed on 362 students selected by cluster sampling method among students of Mazandaran University. The Cronbach’s alpha coefficient was used to check the reliability of the questionnaire and the confirmatory factor analysis was used to determine the factor validity. According to [15] and [14], the results showed that this questionnaire had acceptable internal consistency and Cronbach’s alpha coefficient in the subtest. It ranged from 0.71 to 0.78. Also, the results of the confirmatory factor analysis confirm that the structure of the questionnaire has an acceptable fit to the data and all the goodness of fit indices confirm the model. Therefore, the questionnaire can be a useful tool for evaluating students’ perceptions of assessment and evaluation tasks.

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Copula functions are  useful tools in modeling the dependence between random variables, but most existing copula functions are symmetric, while in many applications, asymmetric joint functions are required. One of these applications is reliability modeling, where asymmetric joint functions can explain different tail dependencies and provide a better model. Therefore, the theory of constructing asymmetric copula functions that can model a wider range of data has been developed. In this research, while reviewing the methods of creating asymmetric copula functions that can provide various tail dependencies, these functions are used to estimate the two-dimensional reliability of data on the age an usage of Rana and Dana cars.

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In this article, the number of failures of a coherent system has been studied under the assumption that the lifetime of system components are non-distributed discrete and dependent random variables. First, the probability that exactly

i

Failure
i=0, ..., n-k,

in a system

$k$

From

n

Under the condition that the system at the time of monitoring

t

it works

it will be counted. In the following, this result has been generalized to other coherent systems. In addition, it has been shown that in the case of independence and co-distribution of component lifetimes, the probability obtained is consistent with the corresponding probability in the continuous state obtained in the existing literature. Finally, by presenting practical examples, the behavior of this probability has been investigated in the case that the system components have interchangeable and necessarily non-distributed lifetimes