PropertiesTableLogNormal.tex

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\begin{table*}[t!]
\caption[Log-normal distribution -- Properties]{Properties of the log-normal distribution}
 \begin{align*}
\text{\hyperref[PropertiesSec]{Properties}}  \quad& \\
\text{notation} \quad & \op{LogNormal}(x\given a, \vartheta,\beta ) \checked  	
\\
\text{PDF}\quad &   \frac{|\beta|}{\sqrt{2\pi \vartheta^2}} \Left(\frac{x-a}{\vartheta}\Right)^{-1} \exp\Left\{-\frac{1}{2} \Left( \beta \ln \frac{x-a}{\vartheta} \Right)^2 \Right\}	\checked					
\\
\text{CDF} \big/ \text{CCDF} \quad  &   \tfrac{1}{2} +  \tfrac{1}{2}\text{erf}\Left( \sfrac{1}{\sqrt{2}} \beta \ln \frac{x-a}{\vartheta} \Right) \checked
& \hspace{-2em} \vartheta >0 \ \big/ \ \vartheta <0
\\
\text{parameters}\quad &   a,\  \vartheta,\ \beta \text{ in } \Real		 \checked
\\
\text{support} \quad &   x \in [a, +\infty] \quad \vartheta>0 \checked
\\
& x \in [-\infty,a] \quad \vartheta<0 \checked
\\
\text{median} \quad  &  a+ \vartheta \checked
\\
\text{mode} \quad  & a+ \vartheta e^{-\beta^{-2}} \checked
\\
\text{mean} \quad  &  a + \vartheta e^{\tfrac{1}{2} \beta^{-2}} \checked
\\
\text{variance} \quad  & \vartheta^2 (e^{\beta^{-2}} -1) e^{\beta^{-2}} \checked
\\
\text{skew} \quad  & \op{sgn}(\vartheta)\  ( e^{\beta^{-2}} +2) \sqrt{e^{\beta^{-2}} -1} \checked
\\
\text{ex. kurtosis} \quad  &  e^{4\beta^{-2}} +2e^{3\beta^{-2}} +3e^{2\beta^{-2}} -6 \checked
\\
\text{entropy} \quad  & \tfrac{1}{2} + \tfrac{1}{2} \ln(2\pi\beta^{-2}) + \ln | \vartheta | \checked
\\
\text{MGF} \quad  &  \text{doesn't exist in general} \checked
\\
\text{CF} \quad  &  \text{no simple closed form expression} \checked
\end{align*}
\end{table*}