PropertiesTablePowerFunc.tex

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\begin{table*}[t!]
\caption[Power function distribution -- Properties]{Properties of the power function distribution}
\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation}\quad &   \text{PowerFn}(x\given a,s,\beta) 
\\
\text{PDF}\quad &   \Left|\frac{\beta}{ s}\Right|\Left(\frac{x-a}{s} \Right)^{\beta -1}  \checked
\\
 \text{CDF}  \big/ \text{CCDF} \quad  &   \Left(\frac{x-a}{s} \Right)^{\beta } & \tfrac{s}{\beta}  >0 \ \big{/} \ \tfrac{s}{\beta}<0  \checked
\\
\text{parameters}\quad &   a, s, \beta \text{ in } \Real
\\
\text{support} \quad &    x \in [a,a+s] & s>0,\ \beta >0 \checked
 \\ 			 &  x\in[a+s,a] &s<0,\ \beta >0   \checked
 \\  			 &  x\in[a+s,+\infty]& s>0,\ \beta<0 \checked
 \\  			&  x\in[-\infty,a+s] & s<0,\ \beta<0  \checked
\\
%\text{median} \quad  &  \cdots
%\\
\text{mode} \quad  & a & \beta >0 \checked
\\
& a+s & \beta <0  \checked
\\
\text{mean} \quad  &a+  \frac{s\beta}{\beta+1} &  \beta \notin [-1,0] \checked
\\
\text{variance} \quad  & \frac{ s^2 \beta }{ (\beta+1)^2 (\beta+2) } & \beta \notin [-2,0] \checked
\\
\text{skew} \quad  & \op{sgn}(\tfrac{\beta}{s})\ \frac{2(1-\beta)}{(\beta+3)}  \sqrt{\frac{\beta+2}{\beta}}
& \beta \notin [-3,0] 
\checked
\\
\text{ex. kurtosis} \quad  &  \frac{6(\beta^3-\beta ^2- 6 \beta +2)}{\beta(\beta +3)(\beta +4)} & \beta \notin [-4,0] \checked
\\
%\text{entropy} \quad  &  \ldots
%\\
\text{MGF} \quad  &  \text{undefined}
%\\
%\text{CF} \quad  & \ldots
\end{align*}
\end{table*}