PropertiesTableGammaExp.tex

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\begin{table*}[tp!]
\caption[Gamma-exponential distribution -- Properties]{Properties of the gamma-exponential distribution}
\begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad &  \op{GammaExp}(x \given \nu, \lambda, \alpha)  			\checked
\\
\text{PDF}\quad &   \frac{1}{ \Gamma(\alpha) |\lambda|}  \exp\Left\{- \alpha \Left(\frac{x-\nu}{\lambda}\Right) - \exp\Left(-\frac{x-\nu}{\lambda}\Right)  \Right\} \checked \hspace{-8em}												
\\
\text{CDF} \big/ \text{CCDF} \quad  &   Q\Left(\alpha, e^{-\frac{x-\nu}{\lambda} }\Right) 
& \lambda>0 \ \big/ \ \lambda<0  \checked
\\
\text{parameters}\quad &   \nu,\ \lambda,\ \alpha,\   \text{in } \Real, \ \alpha>0, \ 		\checked
\\
\text{support} \quad &   x \in [-\infty, +\infty]									\checked
\\
\text{mode} \quad  & \nu -\lambda \ln \alpha \checked
\\
\text{mean} \quad  &   \nu- \lambda \psi(\alpha)	\checked
\\
\text{variance} \quad  &  \lambda^2 \psi_1(\alpha)	\checked
\\
\text{skew} \quad  &   -\text{sgn}(\lambda) \tfrac{\psi_2(\alpha) }{  \psi_1(\alpha)^{3/2} }  \checked
\\
\text{ex. kurtosis} \quad  &     \tfrac{ \psi_3(\alpha) }{  \psi_1(\alpha)^{2}} \checked
\\
%\text{entropy} \quad  &  \ln \Gamma(\alpha) |\lambda|  - \alpha \psi(\alpha) + \alpha
%\\
\text{MGF} \quad  &   e^{\nu t } \frac{\Gamma(\alpha-\lambda t)}{\Gamma(\alpha)}  \checked
& \text{\cite{Johnson1995}}  %after Eq 22.227
\\
\text{CF} \quad  &  e^{i \nu t } \frac{\Gamma(\alpha-i \lambda t)}{\Gamma(\alpha)} \checked
% Johnson1995, Eq 22.211, modulo typos
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