\frac{3}{2},\alpha x^{2\beta}\right) }~~~~~  ;x> t$$ The corresponding probability density function is $$f_{R_t}(x)= \dfrac{4 \alpha^{\frac{3}{2}}\beta (x+t)^{3\beta-1}e^{-\alpha (x+t)^{2\beta}}}{\sqrt{\pi}-2\gamma\left( \frac{3}{2},\alpha x^{2\beta}\right) }$$ Thus, hazard function is obtained as $$h_{R_t}(x)=\dfrac{4 \alpha^{\frac{3}{2}}\beta (x+t)^{3\beta-1}e^{-\alpha (x+t)^{2\beta}}}{\sqrt{\pi}-2\gamma\left[ \frac{3}{2},\alpha (x+t)^{2\beta}\right]}$$

-   **Reversed residual lifetime function**\
    The survival function for MOEIED is given by $$S_{\bar{R}_t}=\dfrac{F(t-x)}{F(t)}= \dfrac{\gamma\left[ \frac{3}{2},\alpha (t-x)^{2\beta}\right] }{\gamma\left( \frac{3}{2},\alpha x^{2\beta}\right) }~~~~~     ;0\le x<t$$ The associated pdf is evaluated as; $$f_{\bar{R}_t}=\dfrac{2 \alpha^{\frac{3}{2}}\beta (t-x)^{3\beta-1}e^{-\alpha (t-x)^{2\beta}}}{\gamma\left( \frac{3}{2},\alpha x^{2\beta}\right) }$$ Hence, the hazard function based on reversed residual lifetime is obtained as $$h_{\bar{R}_t}=\dfrac{2 \alpha^{\frac{3}{2}}\beta (t-x)^{3\beta-1}e^{-\alpha (t-x)^{2\beta}}}{\gamma\left[ \frac{3}{2},\alpha (t-x)^{2\beta}\right]}$$

Entropy Measurements
====================

In information theory, entropy measurement plays a vital role to study the uncertainty associated with the probability distribution. In this section, we discuss the different measure of change. For more detail about entropy measurement, see Reniyi (1961).

Renyi Entropy
-------------

Renyi entropy of a r.v. $x$ is defined as $$\begin{split}
R_E&=\dfrac{1}{(1-\in)}\ln\left[ \int_{x=0}^{\infty}f_{w}^{\in}(x,\alpha,\beta)dx\right]\\&=\dfrac{1}{(1-\in)}\ln \left[ \int_{x=0}^{\infty}\left\lbrace \dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta x^{3\beta-1}e^{-\alpha x^{2\beta}} \right\rbrace^{\in} dx \right] 
\end{split}$$ Hence, after solving the internal, we get the following [$$\begin{split}
R_E&=\dfrac{1}{(1-\in)}\left[\lambda \ln 4-\frac{\lambda}{2}\ln\pi+\lambda\ln\beta-\dfrac{1-\lambda-2\beta}{2\beta}\ln\alpha-\dfrac{3\lambda\beta-\lambda+1}{2\beta}\ln \lambda+\ln \left( \dfrac{3\beta\lambda-\lambda+1}{2\beta}\right)  \right] 
\end{split}$$]{}

$\Delta$-Entropy
----------------

The $\beta$-entropy is obtained as follows $$\Delta_E=\dfrac{1}{\Delta-1}\left[1-\int_{x=0}^{\infty}f^{\Delta}(x,\alpha,\beta)dx \right]$$ Using pdf (1.4) and after simplification the expression for $\beta$-entropy is given by; $$\Delta_E=\dfrac{1}{\Delta-1}\left[1-\left( \dfrac{4}{\sqrt{\pi}}\right) ^{\Delta}\beta^{\Delta} \left( \dfrac{1}{\alpha}\right)^{\dfrac{1-\Delta-2\beta}{2\beta}} \left( \dfrac{\Gamma\left(\dfrac{3\Delta\beta-\Delta+1}{2\beta} \right) }{\Delta^{\dfrac{3\Delta\beta-\Delta+1}{2\beta}}}\right)  