Consistently use Gamma & Exponential distribution

Textbooks use the beta parameter with different semantics. Some use it
as rate parameter and some as inverse scale. This commit switches
Gamma and Expoential distribution over to the rate parameterization, and
adds a footnote to clarify this notational inconsistency.

Closes #13.

Author: mavam

VERSION

@@ -1 +1 @@
-0.1.3
+0.1.4

probstat.tex

@@ -286,7 +286,7 @@
 % Gamma
 \newcommandx\gam[1][1={\alpha,\beta}]{\textrm{Gamma}\left({#1}\right)}
 \newcommandx\dgamma[3][1=x,2=\alpha,3=\beta]%
-{\frac{1}{\Gamma\left( #2 \right) #3^{#2}} #1^{#2 -1}e^{- #1 / #3}}
+{\frac{#3^{#2}}{\Gamma\left( #2 \right)} #1^{#2-1}e^{-#3#1}}
 
 % InverseGamma
 \newcommandx\invgamma[1][1={\alpha,\beta}]{\textrm{InvGamma}\left({#1}\right)}

stat-cookbook.tex

@@ -25,6 +25,7 @@
 \usepackage{multicol}
 \usepackage{rotating}
 \usepackage{tikz}
+\usepackage{threeparttable}
 \usepackage{url}
 \usepackage{xspace}
 
@@ -185,7 +186,7 @@ \subsection{Discrete Distributions}
 
 \subsection{Continuous Distributions}
 
-\begin{center}
+\begin{threeparttable}
 \small
 %\newcolumntype{L}{>{\varwidth[c]{\linewidth}}l<{\endvarwidth}}
 \newcolumntype{M}{>{\begin{math}\displaystyle}c<{\end{math}}}
@@ -244,14 +245,14 @@ \subsection{Continuous Distributions}
   & \frac{2d_2^2(d_1+d_2-2)}{d_1(d_2-2)^2(d_2-4)} %\; d_2 > 4
   & \\[3ex]
 
-  Exponential & \ex & \pex & \dex &
+  Exponential\tnote{$\ast$} & \ex & \pex & \dex &
   \beta & \beta^2 &
-  \frac{1}{1-\beta s} \left(s < 1/\beta \right) \\[3ex]
+  \frac{1}{1-\frac{s}{\beta}} \left(s<\beta\right) \\[3ex]
 
-  Gamma & \gam &
-  \frac{\gamma(\alpha,x/\beta)}{\Gamma(\alpha)} & \dgamma &
-  \alpha\beta & \alpha\beta^2 &
-  \left( \frac{1}{1-\beta s} \right)^\alpha \left( s < 1/\beta \right)\\[3ex]
+  Gamma\tnote{$\ast$} & \gam &
+  \frac{\gamma(\alpha,\beta x)}{\Gamma(\alpha)} & \dgamma &
+  \frac{\alpha}{\beta} & \frac{\alpha}{\beta^2} &
+  \left(\frac{1}{1-\frac{s}{\beta}} \right)^\alpha \left(s<\beta\right)\\[3ex]
 
   Inverse Gamma & \invgamma & \pinvgamma & \dinvgamma &
   \frac{\beta}{\alpha-1} \; \alpha>1 &
@@ -284,7 +285,12 @@ \subsection{Continuous Distributions}
 
   \bottomrule
 \end{tabular}
-\end{center}
+\begin{tablenotes}
+\item[$\ast$] We use the \emph{rate} parameterization where
+  $\beta=\frac{1}{\lambda}$. Some textbooks use $\beta$ as \emph{scale}
+  parameter instead~\cite{Wasserman03}.
+\end{tablenotes}
+\end{threeparttable}
 
 \begin{figure}[H]
   \includegraphics[scale=0.35]{figs/uniform-pdf.pdf}