For some other details on the exponential distribution, readers can go through Oguntunde and Adejumo a. To derive the cdf of the Weibull exponential distribution, Eq. It can be observed in Fig. A plot for the cdf of the proposed model at some selected parameter values is as shown in Fig. Some basic properties of the Weibull exponential distribution: In this sub-section, expressions for some basic mathematical properties of the proposed Weibull exponential distribution are provided.

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For some other details on the exponential distribution, readers can go through Oguntunde and Adejumo a. To derive the cdf of the Weibull exponential distribution, Eq.

It can be observed in Fig. A plot for the cdf of the proposed model at some selected parameter values is as shown in Fig. Some basic properties of the Weibull exponential distribution: In this sub-section, expressions for some basic mathematical properties of the proposed Weibull exponential distribution are provided. Therefore, the reliability function of the proposed model is given by Eq.

The corresponding plot for the reliability function of the Wei-exponential distribution is as shown in Fig. The failure rate is mathematically given by: Therefore, the corresponding failure rate for the Weibull exponential distribution is expressed as in Eq. Following Bourguignon et al. With this understanding, the failure rate of the Weibull exponential distribution is further expressed as Eq.

In other words: Proof: Also: This result shows that the proposed model has at least a unique mode. Moments: Bourguignon et al. Therefore, the rth moment of X is given by Eq. Therefore, the pdf of the ith order statistic for a random sample X1, ,Xn from the Weibull exponential distribution is given as: 26 Following Bourguignon et al.

By the method of Maximum Likelihood Estimation MLE , the likelihood function is given by: The log-likelihood function, l is expressed as: 30 The solution of the non-linear system of equations obtained by differentiating Eq.

The solution can also be obtained directly by using R software when data sets are available. RESULTS To assess the flexibility of the Weibull exponential distribution over the well-known exponential distribution, two real data sets are used and analyses performed with the aid of R software.

Data set I: The first data is on the breaking stress of carbon fibres of 50 mm length GPa. The data is as follows: The summary of the data is given in Table 1. Data set II: The second data set is on the strengths of 1. The data was originally obtained by workers at the UK National Physical Laboratory and it has been used by Smith and Naylor and Bourguignon et al. The data is as follows: The summary of the data is given in Table 3.

Readers can also go through Aryal and Tsokos , Bourguignon et al. Table 1: Data summary for breaking stress of carbon fibres Table 2: Performance of the distributions Standard errors in parentheses AIC: Alkaike information criteria Data summary for strength of 1. In Table 4 , the AIC corresponding to the Weibull exponential distribution is lower than that of the exponential distribution.

Hence, the Weibull exponential distribution performed better than the exponential distribution. This result supports the claim in section 1 of this article that "Generalizing standard or baseline distributions has produced several compound distributions that are more flexible compared to the baseline distributions". This can also be justified by the works of some other notable authors; For instance, Aryal and Tsokos where, Transmuted Weibull distribution performed better than the Weibull distribution, Jafari et al.

It is also good to note that reverse might be the case in some situations, for instance, a generalization proposed in Oguntunde and Adejumo b , the so called generalized inverted generalized exponential distribution did not perform better than a sub-model proposed by the same author inverted generalized exponential distribution based on the data set used.

The shape of model could be unimodal or decreasing depending on the value of the parameters. Some basic mathematical properties of proposed model are rigorously discussed. The Weibull exponential distribution is useful as a life testing model. The model is applied to two real life data sets and it can be said that the Weibull exponential distribution is more flexible than the exponential distribution. Lee and F. Famoye, Gumbel-weibull distribution: Properties and applications.

Applied Stat. Methods,

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## EXPONENTIATED WEIBULL DISTRIBUTION PDF

Malacage Right click equation to reveal menu options. Express 22 17 The Exponentiated Weibull distribution The Exponentiated Weibull distribution is a generalisation of the Weibull distribution which is obtained by exponentiating the Weibull exponetniated distribution function. March 28, Published: From Wikipedia, the free encyclopedia. The Gamma-Gamma GG model is shown for comparison purposes. The Exponentiated Weibull distribution Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipfâ€”Mandelbrot.

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