$$\begin{split}
MD&=\int_{x}|x-\mu| f(x,\alpha,\beta) dx\\&=\int_{x=0}^{\mu}(\mu-x)f(x,\alpha,\beta) dx+\int_{x=\mu}^{\infty}(x-\mu) f(x,\alpha,\beta) dx
\end{split}$$ After simplification, we get $$\begin{split}
MD&=2\mu F(\mu)-2\mu+2\int_{\mu}^{\infty} f(x,\alpha,\beta) dx\\&=(\mu-1)\left[ \dfrac{4}{\sqrt{\pi}}\gamma\left( \frac{3}{2},\alpha \mu^{2\beta}\right)-2\right] 
\end{split}$$

Generating Functions
--------------------

In distribution theory, the role of generating functions are very usefull to generate the respective moments of the distribution and also these functions are uniquely determine the distribution. The different generating function for PMaD $(\alpha,\beta)$ have been caluculated as follows;

-   Moment generating function (mgf) $M_X(t)$ for a random variable $X$ is obatined as $$M_X(t)=E(e^{tx})=\dfrac{2}{\sqrt{\pi}}\sum_{i=0}^{\infty}\frac{1}{j!} \left( \dfrac{t}{\alpha^{2\beta}}\right)^{r}  \Gamma\left( \dfrac{3\beta+r}{2\beta}\right)$$

-   Characteristics function (chf) $\phi_X(t)$ for random variable $X$ is obtained by replacing $t$ by $jt$, $$\phi_X(t)=E(e^{jtx})=\dfrac{2}{\sqrt{\pi}}\sum_{i=0}^{\infty}\frac{1}{j!} \left( \dfrac{jt}{\alpha^{2\beta}}\right)^{r}  \Gamma\left( \dfrac{3\beta+r}{2\beta}\right)$$ where, $j^2=-1$.

-   The kumulants generating function (KGF) is obtained as $$K_X(t)=\ln \left[ \dfrac{2}{\sqrt{\pi}}\sum_{i=0}^{\infty}\frac{1}{j!} \left( \dfrac{t}{\alpha^{2\beta}}\right)^{r}  \Gamma\left( \dfrac{3\beta+r}{2\beta}\right)\right]$$

Conditional Moment and MGF
--------------------------

Let $X$ be a random variable from PMaD$ (\alpha,\beta)$, then conditional moments $E(X^r|X>k)$ and conditional mgf $E(e^{tx}|X>k)$ are evaluated in following expressions; $$\begin{split}
E(X^r|X>k)&=\dfrac{\int_{x>k}x^r f(x,\alpha,\beta) dx}{\int_{x>k} f(x,\alpha,\beta) dx}\\&=\dfrac{2\left(\frac{1}{\alpha}\right)^{\frac{r}{2\beta}}\gamma\left( \dfrac{3\beta+r}{2\beta},\alpha x^{2\beta}\right)}{\sqrt{\pi}-2\gamma\left( \frac{3}{2},\alpha x^{2\beta}\right)}
\end{split}$$ $$\begin{split}
E(e^{tx}|X>k)&=\dfrac{\int_{x>k}e^{tx}f(x,\alpha,\beta) dx}{\int_{x>k} f(x,\alpha,\beta) dx}\\&=\dfrac{2\sum_{i=0}^{\infty}\dfrac{t^i}{i!}\left(\frac{1}{\alpha}\right)^{\frac{r}{2\beta}}\gamma\left( \dfrac{3\beta+r}{2\beta},\alpha x^{2\beta}\right)}{\sqrt{\pi}-2\gamma\left( \frac{3}{2},\alpha x^{2\beta}\right)}
\end{split}$$ respectively.

Bonferroni and Lorenz Curves
----------------------------

In economics to measure the income and poverty level, Bonferroni and Lorenz curves are frequently used. These two have good linkup to each other 