A New Distribution with two parameters to Lifetime Data

In this paper, we proposed a new distribution to lifetime data with two parameters, the proposed distribution have increasing, decreasing and unimodal failure rates function. Some mathematical properties of the new distribution, including hazard function, moments, Estimation of Reliability, distribution of the order statistics and observed information matrix were presented. To estimate the model parameters, the Maximum Likelihood Estimate (MLE) technique was utilized. Then, one real data set were applied to show the significance and flexibility of the new distribution.

In this section, we introduce some lifetime distributions proposed by scientists for modeling such data types.
The aim of this paper is to introduce a new distribution to lifetime data with two parameters and show that it is better model for reliability analysis. Finally, one real data set were applied to show the significance and flexibility of the new distribution. The rest of the paper was organized as follows: A definition was given in Section 2, the moments considered in Section 3. In Section 4 the order statistics is obtain, Estimation of Reliability in Section 5 is obtain, estimations of the parameters were discussed in Section 6, analyses of one real data set was followed in Section 7; and finally, a comparison of the results with the existing distributions and conclusion were presented in Section 8.

A New Distribution
A new distribution was defined using the 2 parameters of α and β through the following Cumulative Distribution Function The PDF is as follows: The new distribution was observed to have the decreasing and unimodal PDFs. Some density curves of the new distribution plotted for varied choices of the parameters are shown in Figure 1. The survival function of the new distribution is given by the following formula: The hazard function is displayed as follows: The new distribution depicts the decreasing, increasing and unimodal failure rates. ; , Letting t x β = we have

Order statistics
cdf and pdf as defined in 1 and 2, respectively. We now give the density of the ith order statistic : ,

Estimation of reliability R for new distribution
Let x f and y f be the pdfs of the independent random variables X and , Y respectively. Then, the reliability function is defined as

Conclusion
In this article, a new distribution with two parameters to Lifetime Data was introduced, which was much more flexible than the Exponentiated Weibull (EW), Modified Weibull (MW), extended generalized Lindley distribution(EGL), entezar distribution(ENT), exponential(E), generalized Lindley (GL), power Lindley(PL), Lindley(L), generalized gamma(GG),Weibull(M), inverse Lindley(IL), Exponential Flexible Weibull Extension (EFWE) and gamma(G) distributions. The distribution proposed in this paper had increasing, decreasing and unimodal hazard rate functions with decreasing and unimodal PDF distributions. A mathematical treatment of this distribution consisting of the moments and asymptotic properties of the MLEs of the obtained unknown parameters was provided. Ultimately, application of the new distribution to one real data set was performed to illustrate the better fit of this distribution compared to the other models.