# Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
## Appendix E The First Raw and Central Moments of IPV Distribution  
To determine the raw moments of the Pearson V (IPV) distribution, the following substitutions in the expression for the probability density function are required:  
\[f(x)=\frac{\exp\left(-\frac{\beta}{x-\gamma}\right)}{\beta\cdot\Gamma(\alpha-1)} \cdot\left(\frac{x-\gamma}{\beta}\right)^{-\alpha}\]  
It is noted:  
\[t=\frac{\beta}{x-\gamma}\quad x=\gamma+\frac{\beta}{t}\quad dx=-\frac{\beta}{t ^{2}}dt\]  
Thus, the first four raw moments are:  
\[m_{1}^{\prime}=\int\limits_{\gamma}^{\infty}x\cdot f(x)=\int \limits_{\gamma}^{\infty}x\cdot\frac{\exp\left(-\frac{\beta}{x-\gamma}\right)} {\beta\cdot\Gamma(\alpha-1)}\cdot\left(\frac{x-\gamma}{\beta}\right)^{-\alpha} dx=\frac{1}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-\left(\gamma+\frac{ \beta}{t}\right)t^{\alpha-2}\cdot e^{-t}\cdot dt=\] \[\frac{\gamma}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-t^{ \alpha-2}\cdot e^{-t}\cdot dt+\frac{\beta}{\Gamma(\alpha-1)}\cdot\int\limits_{ \infty}^{0}-t^{\alpha-3}\cdot e^{-t}\cdot dt=\gamma+\frac{\beta}{\Gamma(\alpha -1)}\cdot\Gamma(\alpha-2)\]  
\[m_{2}^{\prime}=\int\limits_{\gamma}^{\infty}x^{2}\cdot f(x)=\int\limits_{\gamma }^{\infty}x\cdot\frac{\exp\left(-\frac{\beta}{x-\gamma}\right)}{\beta\cdot \Gamma(\alpha-1)}\cdot\left(\frac{x-\gamma}{\beta}\right)^{-\alpha}dx=\frac{1} {\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-\left(\gamma+\frac{\beta}{t} \right)^{2}\cdot t^{\alpha-2}\cdot e^{-t}\cdot dt=\]  
\[\frac{\gamma^{2}}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-t^{\alpha-2} \cdot e^{-t}\cdot dt+\frac{2\cdot\gamma\cdot\beta}{\Gamma(\alpha-1)}\cdot\int \limits_{\infty}^{0}-t^{\alpha-3}\cdot e^{-t}\cdot dt+\frac{\beta^{2}}{\Gamma( \alpha-1)}\cdot\int\limits_{0}^{\infty}-t^{\alpha-4}\cdot e^{-t}\cdot dt=\]  
\[\gamma^{2}+\frac{2\cdot\gamma\cdot\beta}{\Gamma(\alpha-1)}\cdot\Gamma(\alpha- 2)+\frac{\beta^{2}}{\Gamma(\alpha-1)}\cdot\Gamma(\alpha-3)\]  
\[m_{3}^{\prime}=\int\limits_{\gamma}^{\infty}x^{3}\cdot f(x)=\int\limits_{ \gamma}^{\infty}x\cdot\frac{\exp\left(-\frac{\beta}{x-\gamma}\right)}{\beta \cdot\Gamma(\alpha-1)}\cdot\left(\frac{x-\gamma}{\beta}\right)^{-\alpha}dx= \frac{1}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-\left(\gamma+\frac{ \beta}{t}\right)^{4}\cdot t^{-2}\cdot e^{-t}\cdot dt=\]  
\[\frac{\gamma^{4}}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-t^{\alpha-2} \cdot e^{-t}\cdot dt+\frac{4\cdot\gamma^{2}\cdot\beta}{\Gamma(\alpha-1)}\cdot \int\limits_{\infty}^{0}-t^{\alpha-3}\cdot e^{-t}\cdot dt+\frac{6\cdot\gamma^{2 }\cdot\beta^{2}}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{0}-t^{\alpha-4} \cdot e^{-t}\cdot dt-\]  
\[\frac{4\cdot\gamma\cdot\beta^{3}}{\Gamma(\alpha-1)}\cdot\int\limits_{\infty}^{ -t^{\alpha-5}\cdot e^{-t}\cdot dt+\frac{\beta^{4}}{\Gamma(\alpha-1)}\cdot \int\limits_{\infty}^{0}-t^{\alpha-6}\cdot e^{-t}\cdot dt=\]  
\[\gamma^{4}+\frac{4\cdot\gamma^{3}\cdot\beta}{\Gamma(\alpha-1)}\cdot\Gamma( \alpha-2)+\frac{6\cdot\gamma^{2}\cdot\beta^{2}}{\Gamma(\alpha-1)}\cdot\Gamma( \alpha-3)+\frac{4\cdot\gamma\cdot\beta^{3}}{\Gamma(\alpha-1)}\cdot\Gamma( \alpha-4)+\frac{\beta^{4}}{\Gamma(\alpha-1)}\cdot\Gamma(\alpha-5)\]  
The first four centered moments have the following expressions:  
\[m_{1}=m_{1}^{\prime}=\mu==\gamma+\frac{\beta}{\alpha-1}\]  
\[m_{2}=\sigma^{2}=m_{2}^{\prime}-{m_{1}^{\prime}}^{2}=\frac{\beta^{2}}{(\alpha -3)\cdot(\alpha-2)^{2}}\]  
\[m_{3}=m_{3}^{\prime}-3\cdot m_{2}^{\prime}\cdot m_{1}^{\prime}+2\cdot{m_{1}^{ \prime}}^{3}=\frac{4\cdot\beta^{3}}{(\alpha-4)\cdot(\alpha-3)\cdot(\alpha-2) ^{3}}\]  
\[m_{4}=m_{4}^{\prime}-4\cdot m_{3}^{\prime}\cdot m_{1}^{\prime}+6\cdot{m_{2}^{ \prime}}^{2}-3\cdot{m_{1}^{\prime}}^{4}=\frac{3\cdot\beta^{4}\cdot(\alpha+4)}{( \alpha-5)\cdot(\alpha-4)\cdot(\alpha-3)\cdot(\alpha-2)^{4}}\]Skewness and kurtosis are determined with the following expressions:  
\[\gamma_{1}=C_{s}=\frac{m_{3}}{{m_{2}}^{3/2}}=\frac{4\cdot\sqrt{\alpha-3}}{\alpha-4}\]  
\[\gamma_{2}=C_{k}=\frac{m_{4}}{{m_{2}}^{2}}=\frac{3\cdot(\alpha+4)\cdot(\alpha-3) }{(\alpha-4)\cdot(\alpha-5)}\]