PropertiesTablePearsonVII.tex

% !TEX encoding = UTF-8 Unicode 
% !TEX root = FieldGuide.tex

\begin{table*}[p]
\caption[Pearson VII distribution -- Properties]{Properties of the Pearson VII distribution}
\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad & \text{PearsonVII}(x\given a,s, m) \checked
\\
\text{PDF}\quad &   \frac{1}{|s| B(m-\frac{1}{2}, \frac{1}{2} )} \Left( 1 +\Left( \frac{x-a}{s}\Right)^2 \Right)^{-m} \checked
\\
\text{CDF / CCDF} \quad  &  
     \frac{1}{2} +  \Left(\frac{x-a}{s}\Right) 
     \frac{1}{B(m-\frac{1}{2}, \frac{1}{2})}
     \,_2F_1 \Left ( \frac{1}{2},m;\frac{3}{2}; -\Left(\frac{x-a}{s}\Right)^2 \Right)
     \checked
     % Checked with Wolfram Alpha
     % D[1/2+ (x/s) Hypergeometric2F1[1/2, m , 3/2, -(x/s)^2] / Beta(m-1/2, 1/2), x]
     \hspace{-4em}
     \\
     &
\\
\text{parameters}\quad &   a,\ s,\ m  \in \Real \\ & m>\tfrac{1}{2} \checked
\\
\text{support} \quad &   -\infty < x < +\infty	\checked
\\
\text{median} \quad  &  a					\checked
\\
\text{mode} \quad  & a	\checked
\\
\text{mean} \quad  &  a & m>1	\checked
\\
\text{variance} \quad  & \frac{s^2}{2m-3} & m>\tfrac{3}{2} \checked
\\
\text{skew} \quad  &  0 \checked & m>2 \checked
\\
%\text{ex. kurtosis} \quad  &  \cdots & m>\tfrac{5}{2}  
%\\ 
%\text{entropy} \quad  & \cdots
%\\
\text{MGF} \quad  &  \text{undefined}
\\
\text{CF} \quad  &   e^{iat} \frac{2 K_{m-\half} \Left( s |t|\Right)
                    \cdot \Left(\half s |t| \Right)^{m-\half}}
                    {\Gamma(m-\half)}  & m > \half
                    \checked
\end{align*}
\end{table*}