## 2 The E-WD  
In this section we introduce a new combined statistical model and study its behavior. It is the E-WD with three agency parameters, which also consists of an exponential family and a Weibull distribution. The proposed model is superior to many previous distributions and proves its efficiency in modeling stock movement data in the stock market. The cumulative distribution function (CDF) of the new combined statistical model is as follows:  
\[ \begin{array}{rcl}F(\chi;\psi,\lambda,\theta)&=&\int_{0}^{G(\chi;\lambda, \theta)}f(x;\psi)dx\\ &=&\int_{0}^{1-e^{\lambda\chi^{\theta}}}\psi e^{-x\psi}dx\\ &=&[e^{-x\psi}]_{0}^{1-e^{\lambda\theta}}\\ &=&1-e^{-\left(1-e^{-\lambda\chi^{\theta}}\right)\psi}.\end{array}  \tag{4}\]  
The probability density function (PDF) of the complex statistical model gives as follows:  
\[ \begin{array}{rcl}f(\chi;\psi,\lambda,\theta)&=&\frac{\partial F(\chi;\psi, \lambda,\theta)}{\partial\chi}\\ &=&e^{\left(e^{-\chi^{\theta}\lambda}-1\right)\psi}e^{-\chi^{\theta}\lambda} \chi^{\theta-1}\psi\lambda\theta.\end{array}  \tag{5}\]\(\forall\chi;\lambda,\theta\), and \(\psi\), the function in the aforementioned Equation (5) \(f(\chi;\psi,\lambda,\theta)>0\), but  
\[\int_{0}^{\infty}f(\chi;\psi,\lambda,\theta)d\chi = \int_{0}^{\infty}e^{\left(e^{-\chi^{\theta}\lambda}-1\right)\psi}e^ {-\chi^{\theta}\lambda}\chi^{\theta-1}\psi\lambda\theta d\chi \tag{6}\] \[= \left[-e^{\left(e^{-\chi^{\theta}\lambda}-1\right)}\psi\right]_{0 }^{\infty}\] \[= 1-e^{-\psi}.\]  
The function in Equation (5) is a non-probability PDF because its integral is not equal to one, so the appropriate method to convert it into a probability density function is to multiply it by the reciprocal of integration. The PDF of the proposed E-WD is:  
\[f(\chi;\psi,\lambda,\theta)=\frac{e^{\left(e^{-\chi^{\theta}\lambda}-1\right) \psi-\chi^{\theta}\lambda}\chi^{\theta-1}\psi\lambda\theta}{1-e^{-\psi}},\chi \geq 0;\psi,\lambda,\theta>0. \tag{7}\]  
The corresponding CDF gives as follows:  
\[F(\chi;\psi,\lambda,\theta) = \int_{0}^{\chi}f(u;\psi,\lambda,\theta)d\chi \tag{8}\] \[= \frac{\psi\lambda\theta}{1-e^{-\psi}}\int_{0}^{\chi}e^{\left(-1+ e^{-\psi^{\theta}\lambda}\right)\psi-u^{\theta}\lambda}u^{\theta-1}d\chi\] \[= \frac{1}{1-e^{-\psi}}\left[-e^{\left(-1+e^{-\psi^{\theta}\lambda} \right)}\psi\right]_{0}^{\chi}.\]  
The CDF of E-WD is  
\[F(\chi;\psi,\lambda,\theta)=\frac{1}{1-e^{-\psi}}\left[1-e^{\left(e^{-\chi^{ \theta}\lambda}-1\right)}\psi\right],\chi\geq 0;\psi,\lambda,\theta>0. \tag{9}\]  
Furthermore, the survival function, hazard function, cumulative hazard function, and reverse hazard function of E-WD are given, respectively, by  
\[S(\chi;\psi,\lambda,\theta) = 1-F(\chi;\psi,\lambda,\theta)=\frac{e^{e^{-\chi^{\theta}\lambda }}\psi-1}{e^{\psi}-1}, \tag{10}\] \[h(\chi;\psi,\lambda,\theta) = \frac{f(\chi;\psi,\lambda,\theta)}{S(\chi;\psi,\lambda,\theta)}= \frac{e^{e^{-\chi^{\theta}\lambda}}\psi-\chi^{\theta}\lambda}{e^{e^{-\chi^{ \theta}\lambda}}\psi-1},\] (11) \[= \left[-e^{\left(e^{-\chi^{\theta}\lambda}-1\right)}\psi\right]_{0 }^{\infty}\]0 (12) \[= \left[-e^{\left(e^{-\chi^{\theta}\lambda}-1\right)}\psi\right]_{0 }^{\infty}\]1  
The importance and main motivation for the proposed modification E-WD:  
1. To improve the flexibility and distribution properties of Weibull model. 2. The proposed model can take several forms: a right-skewed form, a left-skewed form, a decreasing form, a curved form, and a symmetric form. 3. A simple way to add an additional parameter that gives an extended distribution with \"heavy tail\" properties and is very useful in modeling stock movement data and financial data. * The important statistical functions of the modified E-WD are presented in closed forms. * The new version has many special statistical features. Now we present the visual representation of the PDF of the E-WD.