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In this paper the exact determination of the distribution of stopping variable, the moment and risk of sequential estimator of the failure rate of exponential distribution, under convex boundary is obtained. The corresponding Poisson Process is used to derive the exact distribution of stopping variable of sequential estimator of the failure rate. In the end the exact values of mean and risk of sequential estimator of the failure rate is given in a table.

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In this manuscript first a brief introduction to the Skew-t and Weighted exponential distributions is considered and some of their important properties will be studied. Then we will show that the Skew-t distribution is prefered to the Weighted exponential distribution in fitting by using the real data. Finally we will prove our claim by using the simulation method.

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in this paper, we discuss generating a random sample from gamma distribution using generalized exponential distribution.

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In this paper, after stating the characteristicof some of continuous distributions including, gamma, Crovelli’s
gamma, Rayleigh, Weibull, Pareto, exponential and generalized gamma distribution with each other,these distributions
were fit on drought data of Guilan state and the best distribution was presented. Then, severity and duration of
the drought of different sites were investigated using standardized precipitation index.

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Tolerance interval is a random interval that contains a proportion of the population with a determined confidence
level and is applied in many application fields such as reliability and quality control. In this paper, based on record
data, we obtain a two-sided tolerance interval for the exponential population. An example of real record data is
presented. Finally, we discuss the accuracy of proposed tolerance intervals through a simulation study.

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Nadarajah and Haghighi (2011) introduced a new generalization of the exponential distribution as an alternative
to the gamma, Weibull and exponeniated exponential distributions. In this paper, a generalization of the Nadarajah–
Haghighi (NH) distribution namely exponentiated generalized NH distribution is introduced and discussed. The
properties and applications of proposed model to real data are discussed. A Monte Carlo simulation experiment will
be conducted to evaluate the maximum likelihood estimators of unknown parameters.

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 ‎Censored samples are discussed in experiments of life-testing; i.e‎. ‎whenever the experimenter does not observe the failure times of all units placed on a life test‎. ‎In recent years‎, ‎inference based on censored sampling is considered‎, ‎so that about the parameters of various distributions such as ‎normal‎, ‎exponential‎, ‎gamma‎, ‎Rayleigh‎, ‎Weibull‎, ‎log normal‎, ‎inverse Gaussian‎, ‎logistic‎, ‎Laplace‎, ‎and Pareto‎, ‎has been inferred based on censored sampling‎.

‎In this paper‎, ‎a procedure for exact hypothesis testing and obtaining confidence interval for mean of the exponential distribution under Type-I progressive hybrid censoring is proposed‎. ‎Then‎, ‎performance of the proposed confidence interval is evaluated using simulation‎. ‎Finally‎, ‎the proposed procedures are performed on a data set‎.

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‎In this study‎, ‎E-Bayesian of parameters of two parameter exponential distribution under squared error loss function is obtained‎. ‎The estimated and the efficiency of the proposed method has been compared with Bayesian estimator using Monte Carlo simulation‎. 

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‎In this paper‎, ‎first‎, ‎we investigate probability density function and the failure rate function of some families of exponential distributions‎. ‎Then we present their features such as expectation‎, ‎variance‎, ‎moments and maximum likelihood estimation and we identify the most flexible distributions according to the figure of probability density function and the failure rate function and finally we offer practical examples of them‎.  

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Life testing often is consuming a very long time for testing. Therefore, the engineers and statisticians are looking for some approaches to reduce the running time. There is a recommended method for reducing the time of failure, such that the stress level of the test units will increase, and then they will fail earlier than normal operating conditions. These approaches are called accelerated life tests. One of the most common tests is called the step stress accelerated life test. In this procedure, the stress applied to the units under the test is increased step by step at a predetermined time. The most important aspect to deal with the step stress model is the optimization of test design. In order to optimize the test plan, the best time to increase the level of stress should be chosen. In this paper, at first the step stress testing described. Then, this test is used for exponential lifetime distribution. Since life data are often not complete, this model is applied to type I censored data. By minimizing the asymptotic variance of the maximum likelihood estimator of reliability at time $xi$, the optimal test plan will be obtained. Finally, the simulation studies and one real data are discussed to illustrate the results. A sensitivity analysis shows that the proposed optimum plan is robust.

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In this study, the interval estimations are prosed for the functions of the parameter in exponential lifetimes, when interval
censoring is used. Optimal monitoring time and simulation studies are examined as well as the applicability of the topics.

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The cumulative distribution and density functions of a product of some random variables following the power distribution with different parameters have been provided.
The corresponding characteristic and moment-generating functions are also derived.
We extend the results to the exponential variables and furthermore, some useful identities have been investigated in detail.

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In risk models, the ruin probabilities and ‎Lundberg‎ bound are calculated despite knowing the statistical distribution of random variables. In the present paper, for collective risk model and discrete time risk model of insurance company for independent and identically distributed claims with light-tailed distribution, the infinite time ruin probabilities are computed using Lundberg bound, moreover the general forms of density functions of random variables of claim sizes are derived. 

‎‎‎For some special cases in the  discrete time risk model, the density functions of claim sizes have the shifted geometric distribution, and for the collective risk model, they always have an exponential distribution.‎ ‎

‎‎Presenting the numerical examples of infinite time ruin probabilities and the simulated values of these probabilities and the Lundberg bound are the final results of this article.

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Risk means a situation in which there is a possibility of deviation from a predicted result. Insurance is one of the methods of risk exposure that leads to the transfer of all or part of the risk from the insured to the insurer. Insurance policies are usually classified into two types: personal and non-life (non-life) insurance. According to this classification, any insurance policy that deals with the health of the insured person or persons is a personal insurance policy, otherwise it will be a nonlife insurance policy. Many insurances in the insurance industry are classified as non-life insurances. Fire, automobile, engineering, shipping, oil and energy insurances are examples of these insurances. Explanation and calculation of three issues in risk models are very important: the ruin probability, the time of ruin and the amount of ruin. In this article, the main and well-known results that have been obtained so far in the field of non-life insurance; Emphasizing the possibility of ruin, it is given along with various examples.