PropertiesTableGamma.tex

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\begin{table*}[tp!]

\caption[Gamma distribution -- Properties]{Properties of the gamma distribution}

\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad &  \op{Gamma}(x\given a, \theta, \alpha) 	\checked
\\
\text{PDF} \quad &
\frac{1}{\Gamma(\alpha)\Left|\theta\Right|} 
\Left(\frac{x-a}{\theta}\Right)^{\alpha  -1}
\exp \Left\{
-  \frac{x-a}{\theta}
\Right\}
\checked \hspace{-8em}						
\\ 
\text{CDF / CCDF } \quad  &    1-Q\Left(\alpha, \tfrac{x - a }{\theta} \Right) \checked
& \theta>0 \, \big/ \,  \theta<0
\\
\text{parameters}\quad &   a,\ \theta,\ \alpha,\  \text{in } \Real, \ \alpha>0
\\
\text{support} \quad &     x \geq a &  \theta > 0
\\
&   x\leq a  &  \theta < 0 
\\
%\text{median} \quad  &  \cdots
%\\
\text{mode} \quad&   a+ \theta (\alpha-1)
& \alpha   \geq 1 \checked
\\ & a & \alpha   \le 1 \checked
\\
\text{mean} \quad& a  + \theta \alpha \checked
\\
\text{variance}  \quad&   \theta^2 \alpha \checked & 
\\
\text{skew} \quad  &  \op{sgn}(\theta)\ \frac{2}{\sqrt{\alpha}}  \checked
\\
\text{ex. kurtosis} \quad  &  \frac{6}{\alpha} \checked
\\
\text{entropy} \quad& 
\ln \bigl(|\theta| \Gamma(\alpha) \bigr)+\alpha + \Left( 1 - \alpha\Right) \psi(\alpha) \checked
\\
\text{MGF} \quad  &   e^{a t} (1- \theta t)^{-\alpha}	\checked
% Wikipedia claims restriction that I don't immediately see in other texts.
\\
\text{CF} \quad  &  e^{i a t} (1- i \theta t)^{-\alpha}		\checked
\end{align*}
\end{table*}