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\section{Overview of all the statistics modules}
All statistics modules require the \eslmod{stats} module, in addition
to the core \eslmod{easel} module.
\vspace{1em}
\begin{tabular}{ll}\hline
\multicolumn{2}{c}{\textbf{Core support:}}\\
\eslmod{stats} & Shared and special functions. \\
\multicolumn{2}{c}{\textbf{Distributions:}}\\
\eslmod{dirichlet} & Dirichlet densities. \\
\eslmod{exponential} & Exponential densities.\\
\eslmod{gamma} & Gamma densities.\\
\eslmod{gev} & Generalized extreme value densities.\\
\eslmod{gumbel} & Gumbel densities.\\
\eslmod{hyperexp} & Hyperexponential densities.\\
\eslmod{mixdchlet} & Mixture Dirichlet densities.\\
\eslmod{mixgev} & Mixtures of generalized extreme value densities.\\
\eslmod{stretchexp} & Stretched exponential densities.\\
\eslmod{weibull} & Weibull densities.\\
\hline
\end{tabular}
\subsection{Available densities and distributions}
Every module implements seven functions:
\begin{tabular}{lll}
\ccode{esl\_*\_pdf} & $P(X=x)$ & probability density function\\
\ccode{esl\_*\_logpdf} & $\log P(X=x)$ & natural log of the PDF \\
\ccode{esl\_*\_cdf} & $P(X \leq x)$ & cumulative distribution function\\
\ccode{esl\_*\_logcdf} & $\log P(X \leq x)$ & natural log of the CDF\\
\ccode{esl\_*\_surv} & $P(X > x)$ & survival function (right tail mass) \\
\ccode{esl\_*\_logsurv} & $\log P(X > x)$ & natural log of the survival function\\
\ccode{esl\_*\_invcdf} & ${ x \mid P(X \leq x) = p }$ & inverse CDF (often useful for sampling)\\
\end{tabular}
\subsubsection{Overview of parameters}
A summary of the parameters of the elemental distributions is as
follows:
\begin{tabular}{lcccc} \hline
\textbf{Distribution}& \textbf{Location} & \textbf{Scale} & \textbf{Shape} & \textbf{PDF} \\\hline
\eslmod{dirichlet} & \multicolumn{3}{c}{ $\alpha_i > 0$, $i=1..K$ } &
$\frac{\Gamma{\sum_i \alpha_i}}{\prod_i \Gamma(\alpha_i)} \prod_i p_i^{\alpha_i-1}$\\
\eslmod{exponential} & $\mu$ & $\lambda$ & - &
$\lambda e^{-\lambda (x - \mu)}$\\
\eslmod{gamma} & $\mu$ & $\lambda$ & $\tau$ &
$ \frac{\lambda^{\tau}}{\Gamma(\tau)} (x-\mu)^{\tau-1} e^{-\lambda (x - \mu)} $\\
\eslmod{gev} & $\mu$ & $\lambda$ & $\alpha$ &
$ \lambda \left[ 1 + \alpha \lambda (x - \mu) \right]^{-\frac{\alpha+1}{\alpha}}
\exp \left\{ - \left[ 1 + \alpha \lambda (x - \mu)
\right]^{-\frac{1}{\alpha}} \right\} $\\
\eslmod{gumbel} & $\mu$ & $\lambda$ & - &
$ \lambda \exp \left[ -\lambda (x - \mu) - e^{- \lambda (x - \mu)} \right] $\\
\eslmod{stretchexp} & $\mu$ & $\lambda$ & $\tau$ &
$ \frac{\lambda \tau}{\Gamma(\frac{1}{\tau})} e^{- [\lambda(x-\mu)]^{\tau}} $\\
\eslmod{weibull} & $\mu$ & $\lambda$ & $\tau$ &
$ \lambda \tau [\lambda(x - \mu)]^{\tau-1} e^{- [\lambda(x-\mu)]^{\tau}}$\\
\hline
\end{tabular}
Additionally, there are \textbf{mixture distributions} composed of a
sum of one of the above elemental densities:
\vspace{1em}
\begin{tabular}{ll} \hline
\textbf{Distribution} & \textbf{PDF}\\ \hline
\eslmod{hyperexp} & $\sum_k q_k P(x \mid \mbox{exponential:}\mu^k,\lambda^k)$\\
\eslmod{mixdchlet} & $\sum_k q_k P(\vec{p} \mid \mbox{Dirichlet:}\vec{\alpha}^k)$\\
\eslmod{mixgev} & $\sum_k q_k P(x \mid \mbox{GEV:}\mu^k,\lambda^k,\alpha^k)$\\
\hline
\end{tabular}
\subsubsection{Dynamic range}
\subsection{Using histograms}
\subsection{Parameter fitting}
\subsubsection{Complete data}
\subsubsection{Binned data}
\subsubsection{Censored data}
\subsubsection{Truncated data}
\subsection{Sampling}
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