PropertiesTableBeta.tex

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\begin{table*}[tp]
\caption[Beta distribution -- Properties]{Properties of the beta distribution}
 \begin{align*}
 \text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{name} \quad & \op{Beta}(x\given a,s,\alpha, \gamma) 	\checked
\\
\text{PDF}\quad &    \frac{1}{B(\alpha, \gamma)}\frac{1}{ \Left|s\Right|}
\Left(\frac{x-a}{s} \Right)^{\alpha -1} \Left (1-\Left(\frac{x-a}{s}\Right)\Right)^{\gamma -1}	\checked
\hspace{-8em}
\\
\text{CDF / CCDF} \quad  & 
 \frac{B\Left( \alpha,\gamma; \sfrac{x-a}{s} \Right)}{B(\alpha,\gamma)} = I( \alpha,\gamma; \sfrac{x-a}{s})
 \checked
& {s} >0 \,\big/ \, {s} <0
\\ 
\text{parameters}\quad &   a,\ s,\ \alpha,\ \gamma, \text{ in } \Real, \\ &  \alpha,\gamma\geq0	\checked
\\
\text{support} \quad &  a \geq x \geq a+s , s>0 \quad a+s \geq x \geq a , s<0 
%\\
%\text{median} \quad   &  \cdots
\\
\text{mode} \quad  &a + s \frac{\alpha-1}{\alpha+\gamma-2}\checked  & \alpha,\gamma > 1
\\
\text{mean} \quad  &   a + s \frac{\alpha}{\alpha+\gamma}	\checked
\\
\text{variance} \quad   & s^2 \frac{\alpha\gamma}{(\alpha+\gamma)^2(\alpha+\gamma+1)} \checked
\\
\text{skew} \quad  &   \op{sgn}(s)\ \frac{2(\gamma-\alpha) \sqrt{\alpha+\gamma+1} }{(\alpha+\gamma+2)\sqrt{\alpha\gamma} } \checked
\\
\text{ex. kurtosis} \quad  &  6\frac{(\alpha-\gamma)^2(\alpha+\gamma+1) - \alpha\gamma(\alpha+\gamma+2) }{\alpha\gamma(\alpha+\gamma+2)(\alpha+\gamma+3)} \checked
\\
\text{entropy} \quad  &  \ln(|s|) + \ln\bigl(B(\alpha,\gamma)\bigr)-(\alpha-1)\psi(\alpha)
\\ &\quad {}-(\gamma-1)\psi(\gamma)+(\alpha+\gamma-2)\psi(\alpha+\gamma) \checked
\\
\text{MGF} \quad  &  \text{not simple} 
\\
\text{CF} \quad  &  {}_{1}F_{1}(\alpha; \alpha+\gamma ; i t) \checked
%\\
%E(X^h) \quad & 
\end{align*}
\end{table*}