Figure 1. Plots of PDF, CDF and HRF of BL2PFD
Moments about Zero
The rth moments about zero of any distribution is described below
\(\mu_{r}^{{}^{\prime}}=\int_{0}^{\beta}x^{r}f(x)dx\)
By solving we get
\(\mu_{r}^{{}^{\prime}}=\frac{\ a_{j}\ a_{i}\ a_{l}\ \beta^{r}}{B\left(a,b\right)\ (a+l+j)}\)
\begin{equation}
\text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{r}{\gamma}+1)}{\Gamma\left(\frac{r}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\
\end{equation}