$\alpha$ and $\beta$ are the scale and shape parameter of the proposed distribution. The probability density function and cumulative distribution function of the PMaD are given by $$f(x,\alpha,\beta)=\dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta x^{3\beta-1}e^{-\alpha x^{2\beta}}~~~~x\ge 0,\alpha,\beta>0$$ $$F(x,\alpha,\beta)=\dfrac{2}{\sqrt{\pi}}\gamma\left( \frac{3}{2},\alpha x^{2\beta}\right)$$ respectively. The different mathematical and statistical properties such as moments, reliability, hazard, median, mode, the coefficient of variation, mean deviation, conditional moments, Lorentz curve, stochastic ordering, residual life, entropy measurements, of PMaD have been derived in following subsections.

![image](density){width="6.5in" height="4in"}

Asymptotic behaviour
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This subsection, described the symptotic nature of density and survival functions for the proposed distribution. To illustrate assymptoic behaviour, at first, we will show that $\lim\limits_{x\rightarrow 0}f(x,\alpha,\beta)=0$ and $\lim\limits_{x\rightarrow \infty}f(x,\alpha,\beta)=0$ . Therefore, using (2.1) $$\begin{split}
\lim\limits_{x\rightarrow 0}f(x,\alpha,\beta)&=\dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta\lim\limits_{x\rightarrow 0} x^{3\beta-1}e^{-\alpha x^{2\beta}}\\&=\dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta\times 0=0
\end{split}$$ $\implies$ $\lim\limits_{x\rightarrow \infty}f(x,\alpha,\beta)=0$\
and $$\begin{split}
\lim\limits_{x\rightarrow \infty}f(x,\alpha,\beta)&=\dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta\lim\limits_{x\rightarrow \infty} x^{3\beta-1}\lim\limits_{x\rightarrow \infty}e^{-\alpha x^{2\beta}}\\&=\dfrac{4}{\sqrt{\pi}} \alpha^{\frac{3}{2}}\beta\times \infty\times 0=0
\end{split}$$ $\implies$ $\lim\limits_{x\rightarrow \infty}f(x,\alpha,\beta)=0$\
Similarly, the asymptotic behaviour of survival function can also be shown and found that $\lim\limits_{x\rightarrow 0}S(x)=1$ and $\lim\limits_{x\rightarrow \infty}S(x)=0$.

Reliability and hazard functions
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The characteristics based on reliability function and hazard function are very useful to study the pattern of any lifetime phenomenon. The reliability and hazard function of the proposed distribution have been derived as;

-   The 