[Scipy-svn] r6462 - trunk/doc/source/tutorial/stats
scipy-svn at scipy.org
scipy-svn at scipy.org
Tue Jun 1 00:09:29 EDT 2010
Author: oliphant
Date: 2010-05-31 23:09:28 -0500 (Mon, 31 May 2010)
New Revision: 6462
Added:
trunk/doc/source/tutorial/stats/continuous.lyx
trunk/doc/source/tutorial/stats/discrete.lyx
Log:
Add back original LyX files
Added: trunk/doc/source/tutorial/stats/continuous.lyx
===================================================================
--- trunk/doc/source/tutorial/stats/continuous.lyx (rev 0)
+++ trunk/doc/source/tutorial/stats/continuous.lyx 2010-06-01 04:09:28 UTC (rev 6462)
@@ -0,0 +1,4972 @@
+#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass article
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry true
+\use_amsmath 2
+\use_esint 0
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 1in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Continuous Statistical Distributions
+\end_layout
+
+\begin_layout Section
+Overview
+\end_layout
+
+\begin_layout Standard
+All distributions will have location (L) and Scale (S) parameters along
+ with any shape parameters needed, the names for the shape parameters will
+ vary.
+ Standard form for the distributions will be given where
+\begin_inset Formula $L=0.0$
+\end_inset
+
+ and
+\begin_inset Formula $S=1.0.$
+\end_inset
+
+ The nonstandard forms can be obtained for the various functions using (note
+
+\begin_inset Formula $U$
+\end_inset
+
+ is a standard uniform random variate).
+
+\end_layout
+
+\begin_layout Standard
+\align center
+
+\size small
+\begin_inset Tabular
+<lyxtabular version="3" rows="16" columns="3">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0pt">
+<column alignment="center" valignment="top" leftline="true" width="0pt">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0pt">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Function Name
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Standard Function
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Transformation
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Cumulative Distribution Function (CDF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $F\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Probability Density Function (PDF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $f\left(x\right)=F^{\prime}\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $f\left(x;L,S\right)=\frac{1}{S}f\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Percent Point Function (PPF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $G\left(q\right)=F^{-1}\left(q\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $G\left(q;L,S\right)=L+SG\left(q\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Probability Sparsity Function (PSF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $g\left(q\right)=G^{\prime}\left(q\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $g\left(q;L,S\right)=Sg\left(q\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Hazard Function (HF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $h_{a}\left(x;L,S\right)=\frac{1}{S}h_{a}\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Cumulative Hazard Functon (CHF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $H_{a}\left(x\right)=$
+\end_inset
+
+
+\begin_inset Formula $\log\frac{1}{1-F\left(x\right)}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $H_{a}\left(x;L,S\right)=H_{a}\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Survival Function (SF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $S\left(x\right)=1-F\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Inverse Survival Function (ISF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Moment Generating Function (MGF)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $M_{Y}\left(t\right)=E\left[e^{Yt}\right]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Random Variates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $Y=G\left(U\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $X=L+SY$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+(Differential) Entropy
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $h\left[X\right]=h\left[Y\right]+\log S$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+(Non-central) Moments
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\mu_{n}^{\prime}=E\left[Y^{n}\right]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $E\left[X^{n}\right]=L^{n}\sum_{k=0}^{N}\left(\begin{array}{c}
+n\\
+k\end{array}\right)\left(\frac{S}{L}\right)^{k}\mu_{k}^{\prime}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Central Moments
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+mean (mode, median), var
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\mu,\,\mu_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $L+S\mu,\, S^{2}\mu_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+skewness, kurtosis
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\gamma_{1}=\frac{\mu_{3}}{\left(\mu_{2}\right)^{3/2}},\,$
+\end_inset
+
+
+\begin_inset Formula $\gamma_{2}=\frac{\mu_{4}}{\left(\mu_{2}\right)^{2}}-3$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\gamma_{1},\,\gamma_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\InsetSpace ~
+
+\end_layout
+
+\begin_layout Subsection
+Moments
+\end_layout
+
+\begin_layout Standard
+Non-central moments are defined using the PDF
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\int_{-\infty}^{\infty}x^{n}f\left(x\right)dx.\]
+
+\end_inset
+
+ Note, that these can always be computed using the PPF.
+ Substitute
+\begin_inset Formula $x=G\left(q\right)$
+\end_inset
+
+ in the above equation and get
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\int_{0}^{1}G^{n}\left(q\right)dq\]
+
+\end_inset
+
+ which may be easier to compute numerically.
+ Note that
+\begin_inset Formula $q=F\left(x\right)$
+\end_inset
+
+ so that
+\begin_inset Formula $dq=f\left(x\right)dx.$
+\end_inset
+
+ Central moments are computed similarly
+\begin_inset Formula $\mu=\mu_{1}^{\prime}$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu_{n} & = & \int_{-\infty}^{\infty}\left(x-\mu\right)^{n}f\left(x\right)dx\\
+ & = & \int_{0}^{1}\left(G\left(q\right)-\mu\right)^{n}dq\\
+ & = & \sum_{k=0}^{n}\left(\begin{array}{c}
+n\\
+k\end{array}\right)\left(-\mu\right)^{k}\mu_{n-k}^{\prime}\end{eqnarray*}
+
+\end_inset
+
+ In particular
+\begin_inset Formula \begin{eqnarray*}
+\mu_{3} & = & \mu_{3}^{\prime}-3\mu\mu_{2}^{\prime}+2\mu^{3}\\
+ & = & \mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}\\
+\mu_{4} & = & \mu_{4}^{\prime}-4\mu\mu_{3}^{\prime}+6\mu^{2}\mu_{2}^{\prime}-3\mu^{4}\\
+ & = & \mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\end{eqnarray*}
+
+\end_inset
+
+ Skewness is defined as
+\begin_inset Formula \[
+\gamma_{1}=\sqrt{\beta_{1}}=\frac{\mu_{3}}{\mu_{2}^{3/2}}\]
+
+\end_inset
+
+ while (Fisher) kurtosis is
+\begin_inset Formula \[
+\gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,\]
+
+\end_inset
+
+ so that a normal distribution has a kurtosis of zero.
+
+\end_layout
+
+\begin_layout Subsection
+Median and mode
+\end_layout
+
+\begin_layout Standard
+The median,
+\begin_inset Formula $m_{n}$
+\end_inset
+
+ is defined as the point at which half of the density is on one side and
+ half on the other.
+ In other words,
+\begin_inset Formula $F\left(m_{n}\right)=\frac{1}{2}$
+\end_inset
+
+ so that
+\begin_inset Formula \[
+m_{n}=G\left(\frac{1}{2}\right).\]
+
+\end_inset
+
+ In addition, the mode,
+\begin_inset Formula $m_{d}$
+\end_inset
+
+, is defined as the value for which the probability density function reaches
+ it's peak
+\begin_inset Formula \[
+m_{d}=\arg\max_{x}f\left(x\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Fitting data
+\end_layout
+
+\begin_layout Standard
+To fit data to a distribution, maximizing the likelihood function is common.
+ Alternatively, some distributions have well-known minimum variance unbiased
+ estimators.
+ These will be chosen by default, but the likelihood function will always
+ be available for minimizing.
+
+\end_layout
+
+\begin_layout Standard
+If
+\begin_inset Formula $f\left(x;\boldsymbol{\theta}\right)$
+\end_inset
+
+ is the PDF of a random-variable where
+\begin_inset Formula $\boldsymbol{\theta}$
+\end_inset
+
+ is a vector of parameters (
+\emph on
+e.g.
+
+\begin_inset Formula $L$
+\end_inset
+
+
+\emph default
+ and
+\begin_inset Formula $S$
+\end_inset
+
+), then for a collection of
+\begin_inset Formula $N$
+\end_inset
+
+ independent samples from this distribution, the joint distribution the
+ random vector
+\begin_inset Formula $\mathbf{x}$
+\end_inset
+
+ is
+\begin_inset Formula \[
+f\left(\mathbf{x};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f\left(x_{i};\boldsymbol{\theta}\right).\]
+
+\end_inset
+
+ The maximum likelihood estimate of the parameters
+\begin_inset Formula $\boldsymbol{\theta}$
+\end_inset
+
+ are the parameters which maximize this function with
+\begin_inset Formula $\mathbf{x}$
+\end_inset
+
+ fixed and given by the data:
+\begin_inset Formula \begin{eqnarray*}
+\boldsymbol{\theta}_{es} & = & \arg\max_{\boldsymbol{\theta}}f\left(\mathbf{x};\boldsymbol{\theta}\right)\\
+ & = & \arg\min_{\boldsymbol{\theta}}l_{\mathbf{x}}\left(\boldsymbol{\theta}\right).\end{eqnarray*}
+
+\end_inset
+
+ Where
+\begin_inset Formula \begin{eqnarray*}
+l_{\mathbf{x}}\left(\boldsymbol{\theta}\right) & = & -\sum_{i=1}^{N}\log f\left(x_{i};\boldsymbol{\theta}\right)\\
+ & = & -N\overline{\log f\left(x_{i};\boldsymbol{\theta}\right)}\end{eqnarray*}
+
+\end_inset
+
+ Note that if
+\begin_inset Formula $\boldsymbol{\theta}$
+\end_inset
+
+ includes only shape parameters, the location and scale-parameters can be
+ fit by replacing
+\begin_inset Formula $x_{i}$
+\end_inset
+
+ with
+\begin_inset Formula $\left(x_{i}-L\right)/S$
+\end_inset
+
+ in the log-likelihood function adding
+\begin_inset Formula $N\log S$
+\end_inset
+
+ and minimizing, thus
+\begin_inset Formula \begin{eqnarray*}
+l_{\mathbf{x}}\left(L,S;\boldsymbol{\theta}\right) & = & N\log S-\sum_{i=1}^{N}\log f\left(\frac{x_{i}-L}{S};\boldsymbol{\theta}\right)\\
+ & = & N\log S+l_{\frac{\mathbf{x}-S}{L}}\left(\boldsymbol{\theta}\right)\end{eqnarray*}
+
+\end_inset
+
+ If desired, sample estimates for
+\begin_inset Formula $L$
+\end_inset
+
+ and
+\begin_inset Formula $S$
+\end_inset
+
+ (not necessarily maximum likelihood estimates) can be obtained from samples
+ estimates of the mean and variance using
+\begin_inset Formula \begin{eqnarray*}
+\hat{S} & = & \sqrt{\frac{\hat{\mu}_{2}}{\mu_{2}}}\\
+\hat{L} & = & \hat{\mu}-\hat{S}\mu\end{eqnarray*}
+
+\end_inset
+
+ where
+\begin_inset Formula $\mu$
+\end_inset
+
+ and
+\begin_inset Formula $\mu_{2}$
+\end_inset
+
+ are assumed known as the mean and variance of the
+\series bold
+untransformed
+\series default
+ distribution (when
+\begin_inset Formula $L=0$
+\end_inset
+
+ and
+\begin_inset Formula $S=1$
+\end_inset
+
+) and
+\begin_inset Formula \begin{eqnarray*}
+\hat{\mu} & = & \frac{1}{N}\sum_{i=1}^{N}x_{i}=\bar{\mathbf{x}}\\
+\hat{\mu}_{2} & = & \frac{1}{N-1}\sum_{i=1}^{N}\left(x_{i}-\hat{\mu}\right)^{2}=\frac{N}{N-1}\overline{\left(\mathbf{x}-\bar{\mathbf{x}}\right)^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Standard notation for mean
+\end_layout
+
+\begin_layout Standard
+We will use
+\begin_inset Formula \[
+\overline{y\left(\mathbf{x}\right)}=\frac{1}{N}\sum_{i=1}^{N}y\left(x_{i}\right)\]
+
+\end_inset
+
+ where
+\begin_inset Formula $N$
+\end_inset
+
+ should be clear from context as the number of samples
+\begin_inset Formula $x_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Alpha
+\end_layout
+
+\begin_layout Standard
+One shape parameters
+\begin_inset Formula $\alpha>0$
+\end_inset
+
+ (paramter
+\begin_inset Formula $\beta$
+\end_inset
+
+ in DATAPLOT is a scale-parameter).
+ Standard form is
+\begin_inset Formula $x>0:$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\alpha\right) & = & \frac{1}{x^{2}\Phi\left(\alpha\right)\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{x}\right)^{2}\right)\\
+F\left(x;\alpha\right) & = & \frac{\Phi\left(\alpha-\frac{1}{x}\right)}{\Phi\left(\alpha\right)}\\
+G\left(q;\alpha\right) & = & \left[\alpha-\Phi^{-1}\left(q\Phi\left(\alpha\right)\right)\right]^{-1}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+M\left(t\right)=\frac{1}{\Phi\left(a\right)\sqrt{2\pi}}\int_{0}^{\infty}\frac{e^{xt}}{x^{2}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{x}\right)^{2}\right)dx\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+No moments?
+\begin_inset Formula \[
+l_{\mathbf{x}}\left(\alpha\right)=N\log\left[\Phi\left(\alpha\right)\sqrt{2\pi}\right]+2N\overline{\log\mathbf{x}}+\frac{N}{2}\alpha^{2}-\alpha\overline{\mathbf{x}^{-1}}+\frac{1}{2}\overline{\mathbf{x}^{-2}}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Anglit
+\end_layout
+
+\begin_layout Standard
+Defined over
+\begin_inset Formula $x\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \sin\left(2x+\frac{\pi}{2}\right)=\cos\left(2x\right)\\
+F\left(x\right) & = & \sin^{2}\left(x+\frac{\pi}{4}\right)\\
+G\left(q\right) & = & \arcsin\left(\sqrt{q}\right)-\frac{\pi}{4}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 0\\
+\mu_{2} & = & \frac{\pi^{2}}{16}-\frac{1}{2}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & -2\frac{\pi^{4}-96}{\left(\pi^{2}-8\right)^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & 1-\log2\\
+ & \approx & 0.30685281944005469058\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+M\left(t\right) & = & \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\cos\left(2x\right)e^{xt}dx\\
+ & = & \frac{4\cosh\left(\frac{\pi t}{4}\right)}{t^{2}+4}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+l_{\mathbf{x}}\left(\cdot\right)=-N\overline{\log\left[\cos\left(2\mathbf{x}\right)\right]}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Arcsine
+\end_layout
+
+\begin_layout Standard
+Defined over
+\begin_inset Formula $x\in\left(0,1\right)$
+\end_inset
+
+.
+ To get the JKB definition put
+\begin_inset Formula $x=\frac{u+1}{2}.$
+\end_inset
+
+ i.e.
+
+\begin_inset Formula $L=-1$
+\end_inset
+
+ and
+\begin_inset Formula $S=2.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{\pi\sqrt{x\left(1-x\right)}}\\
+F\left(x\right) & = & \frac{2}{\pi}\arcsin\left(\sqrt{x}\right)\\
+G\left(q\right) & = & \sin^{2}\left(\frac{\pi}{2}q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=E^{t/2}I_{0}\left(\frac{t}{2}\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu_{n}^{\prime} & = & \frac{1}{\pi}\int_{0}^{1}dx\, x^{n-1/2}\left(1-x\right)^{-1/2}\\
+ & = & \frac{1}{\pi}B\left(\frac{1}{2},n+\frac{1}{2}\right)=\frac{\left(2n-1\right)!!}{2^{n}n!}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{1}{2}\\
+\mu_{2} & = & \frac{1}{8}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & -\frac{3}{2}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]\approx-0.24156447527049044468\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+l_{\mathbf{x}}\left(\cdot\right)=N\log\pi+\frac{N}{2}\overline{\log\mathbf{x}}+\frac{N}{2}\overline{\log\left(1-\mathbf{x}\right)}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Beta
+\end_layout
+
+\begin_layout Standard
+Two shape parameters
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+a,b>0\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,b\right) & = & \frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}x^{a-1}\left(1-x\right)^{b-1}I_{\left(0,1\right)}\left(x\right)\\
+F\left(x;a,b\right) & = & \int_{0}^{x}f\left(y;a,b\right)dy=I\left(x,a,b\right)\\
+G\left(\alpha;a,b\right) & = & I^{-1}\left(\alpha;a,b\right)\\
+M\left(t\right) & = & \frac{\Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}\,_{1}F_{1}\left(a;a+b;t\right)\\
+\mu & = & \frac{a}{a+b}\\
+\mu_{2} & = & \frac{ab\left(a+b+1\right)}{\left(a+b\right)^{2}}\\
+\gamma_{1} & = & 2\frac{b-a}{a+b+2}\sqrt{\frac{a+b+1}{ab}}\\
+\gamma_{2} & = & \frac{6\left(a^{3}+a^{2}\left(1-2b\right)+b^{2}\left(b+1\right)-2ab\left(b+2\right)\right)}{ab\left(a+b+2\right)\left(a+b+3\right)}\\
+m_{d} & = & \frac{\left(a-1\right)}{\left(a+b-2\right)}\, a+b\neq2\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f\left(x;a,1\right)$
+\end_inset
+
+ is also called the Power-function distribution.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+l_{\mathbf{x}}\left(a,b\right)=-N\log\Gamma\left(a+b\right)+N\log\Gamma\left(a\right)+N\log\Gamma\left(b\right)-N\left(a-1\right)\overline{\log\mathbf{x}}-N\left(b-1\right)\overline{\log\left(1-\mathbf{x}\right)}\]
+
+\end_inset
+
+ All of the
+\begin_inset Formula $x_{i}\in\left[0,1\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Beta Prime
+\end_layout
+
+\begin_layout Standard
+Defined over
+\begin_inset Formula $0<x<\infty.$
+\end_inset
+
+
+\begin_inset Formula $\alpha,\beta>0.$
+\end_inset
+
+ (Note the CDF evaluation uses Eq.
+ 3.194.1 on pg.
+ 313 of Gradshteyn & Ryzhik (sixth edition).
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\alpha,\beta\right) & = & \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha-1}\left(1+x\right)^{-\alpha-\beta}\\
+F\left(x;\alpha,\beta\right) & = & \frac{\Gamma\left(\alpha+\beta\right)}{\alpha\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha}\,_{2}F_{1}\left(\alpha+\beta,\alpha;1+\alpha;-x\right)\\
+G\left(q;\alpha,\beta\right) & = & F^{-1}\left(x;\alpha,\beta\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\left\{ \begin{array}{ccc}
+\frac{\Gamma\left(n+\alpha\right)\Gamma\left(\beta-n\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}=\frac{\left(\alpha\right)_{n}}{\left(\beta-n\right)_{n}} & & \beta>n\\
+\infty & & \textrm{otherwise}\end{array}\right.\]
+
+\end_inset
+
+ Therefore,
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{\alpha}{\beta-1}\quad\beta>1\\
+\mu_{2} & = & \frac{\alpha\left(\alpha+1\right)}{\left(\beta-2\right)\left(\beta-1\right)}-\frac{\alpha^{2}}{\left(\beta-1\right)^{2}}\quad\beta>2\\
+\gamma_{1} & = & \frac{\frac{\alpha\left(\alpha+1\right)\left(\alpha+2\right)}{\left(\beta-3\right)\left(\beta-2\right)\left(\beta-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\quad\beta>3\\
+\gamma_{2} & = & \frac{\mu_{4}}{\mu_{2}^{2}}-3\\
+\mu_{4} & = & \frac{\alpha\left(\alpha+1\right)\left(\alpha+2\right)\left(\alpha+3\right)}{\left(\beta-4\right)\left(\beta-3\right)\left(\beta-2\right)\left(\beta-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\quad\beta>4\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Bradford
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+c & > & 0\\
+k & = & \log\left(1+c\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{c}{k\left(1+cx\right)}I_{\left(0,1\right)}\left(x\right)\\
+F\left(x;c\right) & = & \frac{\log\left(1+cx\right)}{k}\\
+G\left(\alpha\; c\right) & = & \frac{\left(1+c\right)^{\alpha}-1}{c}\\
+M\left(t\right) & = & \frac{1}{k}e^{-t/c}\left[\textrm{Ei}\left(t+\frac{t}{c}\right)-\textrm{Ei}\left(\frac{t}{c}\right)\right]\\
+\mu & = & \frac{c-k}{ck}\\
+\mu_{2} & = & \frac{\left(c+2\right)k-2c}{2ck^{2}}\\
+\gamma_{1} & = & \frac{\sqrt{2}\left(12c^{2}-9kc\left(c+2\right)+2k^{2}\left(c\left(c+3\right)+3\right)\right)}{\sqrt{c\left(c\left(k-2\right)+2k\right)}\left(3c\left(k-2\right)+6k\right)}\\
+\gamma_{2} & = & \frac{c^{3}\left(k-3\right)\left(k\left(3k-16\right)+24\right)+12kc^{2}\left(k-4\right)\left(k-3\right)+6ck^{2}\left(3k-14\right)+12k^{3}}{3c\left(c\left(k-2\right)+2k\right)^{2}}\\
+m_{d} & = & 0\\
+m_{n} & = & \sqrt{1+c}-1\end{eqnarray*}
+
+\end_inset
+
+ where
+\begin_inset Formula $\textrm{Ei}\left(\textrm{z}\right)$
+\end_inset
+
+ is the exponential integral function.
+ Also
+\begin_inset Formula \[
+h\left[X\right]=\frac{1}{2}\log\left(1+c\right)-\log\left(\frac{c}{\log\left(1+c\right)}\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Burr
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+c & > & 0\\
+d & > & 0\\
+k & = & \Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+d\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+d\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c,d\right) & = & \frac{cd}{x^{c+1}\left(1+x^{-c}\right)^{d+1}}I_{\left(0,\infty\right)}\left(x\right)\\
+F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-d}\\
+G\left(\alpha;c,d\right) & = & \left(\alpha^{-1/d}-1\right)^{-1/c}\\
+\mu & = & \frac{\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+d\right)}{\Gamma\left(d\right)}\\
+\mu_{2} & = & \frac{k}{\Gamma^{2}\left(d\right)}\\
+\gamma_{1} & = & \frac{1}{\sqrt{k^{3}}}\left[2\Gamma^{3}\left(1-\frac{1}{c}\right)\Gamma^{3}\left(\frac{1}{c}+d\right)+\Gamma^{2}\left(d\right)\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(\frac{3}{c}+d\right)\right.\\
+ & & \left.-3\Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+d\right)\Gamma\left(\frac{2}{c}+d\right)\right]\\
+\gamma_{2} & = & -3+\frac{1}{k^{2}}\left[6\Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+d\right)\Gamma\left(\frac{2}{c}+d\right)\right.\\
+ & & -3\Gamma^{4}\left(1-\frac{1}{c}\right)\Gamma^{4}\left(\frac{1}{c}+d\right)+\Gamma^{3}\left(d\right)\Gamma\left(1-\frac{4}{c}\right)\Gamma\left(\frac{4}{c}+d\right)\\
+ & & \left.-4\Gamma^{2}\left(d\right)\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+d\right)\Gamma\left(\frac{3}{c}+d\right)\right]\\
+m_{d} & = & \left(\frac{cd-1}{c+1}\right)^{1/c}\,\textrm{if }cd>1\,\textrm{otherwise }0\\
+m_{n} & = & \left(2^{1/d}-1\right)^{-1/c}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Cauchy
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{\pi\left(1+x^{2}\right)}\\
+F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\tan^{-1}x\\
+G\left(\alpha\right) & = & \tan\left(\pi\alpha-\frac{\pi}{2}\right)\\
+m_{d} & = & 0\\
+m_{n} & = & 0\end{eqnarray*}
+
+\end_inset
+
+No finite moments.
+ This is the t distribution with one degree of freedom.
+
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(4\pi\right)\\
+ & \approx & 2.5310242469692907930.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Chi
+\end_layout
+
+\begin_layout Standard
+Generated by taking the (positive) square-root of chi-squared variates.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\nu\right) & = & \frac{x^{\nu-1}e^{-x^{2}/2}}{2^{\nu/2-1}\Gamma\left(\frac{\nu}{2}\right)}I_{\left(0,\infty\right)}\left(x\right)\\
+F\left(x;\nu\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x^{2}}{2}\right)\\
+G\left(\alpha;\nu\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\alpha\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\Gamma\left(\frac{v}{2}\right)\,_{1}F_{1}\left(\frac{v}{2};\frac{1}{2};\frac{t^{2}}{2}\right)+\frac{t}{\sqrt{2}}\Gamma\left(\frac{1+\nu}{2}\right)\,_{1}F_{1}\left(\frac{1+\nu}{2};\frac{3}{2};\frac{t^{2}}{2}\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{\sqrt{2}\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)}\\
+\mu_{2} & = & \nu-\mu^{2}\\
+\gamma_{1} & = & \frac{2\mu^{3}+\mu\left(1-2\nu\right)}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{2\nu\left(1-\nu\right)-6\mu^{4}+4\mu^{2}\left(2\nu-1\right)}{\mu_{2}^{2}}\\
+m_{d} & = & \sqrt{\nu-1}\quad\nu\geq1\\
+m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\frac{1}{2}\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Chi-squared
+\end_layout
+
+\begin_layout Standard
+This is the gamma distribution with
+\begin_inset Formula $L=0.0$
+\end_inset
+
+ and
+\begin_inset Formula $S=2.0$
+\end_inset
+
+ and
+\begin_inset Formula $\alpha=\nu/2$
+\end_inset
+
+ where
+\begin_inset Formula $\nu$
+\end_inset
+
+ is called the degrees of freedom.
+ If
+\begin_inset Formula $Z_{1}\ldots Z_{\nu}$
+\end_inset
+
+ are all standard normal distributions, then
+\begin_inset Formula $W=\sum_{k}Z_{k}^{2}$
+\end_inset
+
+ has (standard) chi-square distribution with
+\begin_inset Formula $\nu$
+\end_inset
+
+ degrees of freedom.
+
+\end_layout
+
+\begin_layout Standard
+The standard form (most often used in standard form only) is
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\frac{\nu}{2}\right)}\left(\frac{x}{2}\right)^{\nu/2-1}e^{-x/2}\\
+F\left(x;\alpha\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x}{2}\right)\\
+G\left(q;\alpha\right) & = & 2\Gamma^{-1}\left(\frac{\nu}{2},q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1}{2}-t\right)^{\nu/2}}\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \nu\\
+\mu_{2} & = & 2\nu\\
+\gamma_{1} & = & \frac{2\sqrt{2}}{\sqrt{\nu}}\\
+\gamma_{2} & = & \frac{12}{\nu}\\
+m_{d} & = & \frac{\nu}{2}-1\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Cosine
+\end_layout
+
+\begin_layout Standard
+Approximation to the normal distribution.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{2\pi}\left[1+\cos x\right]I_{\left[-\pi,\pi\right]}\left(x\right)\\
+F\left(x\right) & = & \frac{1}{2\pi}\left[\pi+x+\sin x\right]I_{\left[-\pi,\pi\right]}\left(x\right)+I_{\left(\pi,\infty\right)}\left(x\right)\\
+G\left(\alpha\right) & = & F^{-1}\left(\alpha\right)\\
+M\left(t\right) & = & \frac{\sinh\left(\pi t\right)}{\pi t\left(1+t^{2}\right)}\\
+\mu=m_{d}=m_{n} & = & 0\\
+\mu_{2} & = & \frac{\pi^{2}}{3}-2\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & \frac{-6\left(\pi^{4}-90\right)}{5\left(\pi^{2}-6\right)^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(4\pi\right)-1\\
+ & \approx & 1.5310242469692907930.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Double Gamma
+\end_layout
+
+\begin_layout Standard
+The double gamma is the signed version of the Gamma distribution.
+ For
+\begin_inset Formula $\alpha>0:$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\alpha\right)}\left|x\right|^{\alpha-1}e^{-\left|x\right|}\\
+F\left(x;\alpha\right) & = & \left\{ \begin{array}{ccc}
+\frac{1}{2}-\frac{1}{2}\Gamma\left(\alpha,\left|x\right|\right) & & x\leq0\\
+\frac{1}{2}+\frac{1}{2}\Gamma\left(\alpha,\left|x\right|\right) & & x>0\end{array}\right.\\
+G\left(q;\alpha\right) & = & \left\{ \begin{array}{ccc}
+-\Gamma^{-1}\left(\alpha,\left|2q-1\right|\right) & & q\leq\frac{1}{2}\\
+\Gamma^{-1}\left(\alpha,\left|2q-1\right|\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+M\left(t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\right)^{a}}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu=m_{n} & = & 0\\
+\mu_{2} & = & \alpha\left(\alpha+1\right)\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & \frac{\left(\alpha+2\right)\left(\alpha+3\right)}{\alpha\left(\alpha+1\right)}-3\\
+m_{d} & = & \textrm{NA}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Doubly Non-central F*
+\end_layout
+
+\begin_layout Section
+Doubly Non-central t*
+\end_layout
+
+\begin_layout Section
+Double Weibull
+\end_layout
+
+\begin_layout Standard
+This is a signed form of the Weibull distribution.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{c}{2}\left|x\right|^{c-1}\exp\left(-\left|x\right|^{c}\right)\\
+F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
+\frac{1}{2}\exp\left(-\left|x\right|^{c}\right) & & x\leq0\\
+1-\frac{1}{2}\exp\left(-\left|x\right|^{c}\right) & & x>0\end{array}\right.\\
+G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
+-\log^{1/c}\left(\frac{1}{2q}\right) & & q\leq\frac{1}{2}\\
+\log^{1/c}\left(\frac{1}{2q-1}\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\mu_{n}=\begin{cases}
+\Gamma\left(1+\frac{n}{c}\right) & n\textrm{ even}\\
+0 & n\textrm{ odd}\end{cases}\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+m_{d}=\mu & = & 0\\
+\mu_{2} & = & \Gamma\left(\frac{c+2}{c}\right)\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & \frac{\Gamma\left(1+\frac{4}{c}\right)}{\Gamma^{2}\left(1+\frac{2}{c}\right)}\\
+m_{d} & = & \textrm{NA bimodal}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Erlang
+\end_layout
+
+\begin_layout Standard
+This is just the Gamma distribution with shape parameter
+\begin_inset Formula $\alpha=n$
+\end_inset
+
+ an integer.
+
+\end_layout
+
+\begin_layout Section
+Exponential
+\end_layout
+
+\begin_layout Standard
+This is a special case of the Gamma (and Erlang) distributions with shape
+ parameter
+\begin_inset Formula $\left(\alpha=1\right)$
+\end_inset
+
+ and the same location and scale parameters.
+ The standard form is therefore (
+\begin_inset Formula $x\geq0$
+\end_inset
+
+)
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & e^{-x}\\
+F\left(x\right) & = & \Gamma\left(1,x\right)=1-e^{-x}\\
+G\left(q\right) & = & -\log\left(1-q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\mu_{n}^{\prime}=n!\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+M\left(t\right)=\frac{1}{1-t}\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 1\\
+\mu_{2} & = & 1\\
+\gamma_{1} & = & 2\\
+\gamma_{2} & = & 6\\
+m_{d} & = & 0\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=1.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Exponentiated Weibull
+\end_layout
+
+\begin_layout Standard
+Two positive shape parameters
+\begin_inset Formula $a$
+\end_inset
+
+ and
+\begin_inset Formula $c$
+\end_inset
+
+ and
+\begin_inset Formula $x\in\left(0,\infty\right)$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,c\right) & = & ac\left[1-\exp\left(-x^{c}\right)\right]^{a-1}\exp\left(-x^{c}\right)x^{c-1}\\
+F\left(x;a,c\right) & = & \left[1-\exp\left(-x^{c}\right)\right]^{a}\\
+G\left(q;a,c\right) & = & \left[-\log\left(1-q^{1/a}\right)\right]^{1/c}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Exponential Power
+\end_layout
+
+\begin_layout Standard
+One positive shape parameter
+\begin_inset Formula $b$
+\end_inset
+
+.
+ Defined for
+\begin_inset Formula $x\geq0.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;b\right) & = & ebx^{b-1}\exp\left[x^{b}-e^{x^{b}}\right]\\
+F\left(x;b\right) & = & 1-\exp\left[1-e^{x^{b}}\right]\\
+G\left(q;b\right) & = & \log^{1/b}\left[1-\log\left(1-q\right)\right]\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Fatigue Life (Birnbaum-Sanders)
+\end_layout
+
+\begin_layout Standard
+This distribution's pdf is the average of the inverse-Gaussian
+\begin_inset Formula $\left(\mu=1\right)$
+\end_inset
+
+ and reciprocal inverse-Gaussian pdf
+\begin_inset Formula $\left(\mu=1\right)$
+\end_inset
+
+.
+ We follow the notation of JKB here with
+\begin_inset Formula $\beta=S.$
+\end_inset
+
+ for
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{x+1}{2c\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2xc^{2}}\right)\\
+F\left(x;c\right) & = & \Phi\left(\frac{1}{c}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\right)\\
+G\left(q;c\right) & = & \frac{1}{4}\left[c\Phi^{-1}\left(q\right)+\sqrt{c^{2}\left(\Phi^{-1}\left(q\right)\right)^{2}+4}\right]^{2}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=c\sqrt{2\pi}\exp\left[\frac{1}{c^{2}}\left(1-\sqrt{1-2c^{2}t}\right)\right]\left(1+\frac{1}{\sqrt{1-2c^{2}t}}\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{c^{2}}{2}+1\\
+\mu_{2} & = & c^{2}\left(\frac{5}{4}c^{2}+1\right)\\
+\gamma_{1} & = & \frac{4c\sqrt{11c^{2}+6}}{\left(5c^{2}+4\right)^{3/2}}\\
+\gamma_{2} & = & \frac{6c^{2}\left(93c^{2}+41\right)}{\left(5c^{2}+4\right)^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Fisk (Log Logistic)
+\end_layout
+
+\begin_layout Standard
+Special case of the Burr distribution with
+\begin_inset Formula $d=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+c & > & 0\\
+k & = & \Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+1\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c,d\right) & = & \frac{cx^{c-1}}{\left(1+x^{c}\right)^{2}}I_{\left(0,\infty\right)}\left(x\right)\\
+F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-1}\\
+G\left(\alpha;c,d\right) & = & \left(\alpha^{-1}-1\right)^{-1/c}\\
+\mu & = & \Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\\
+\mu_{2} & = & k\\
+\gamma_{1} & = & \frac{1}{\sqrt{k^{3}}}\left[2\Gamma^{3}\left(1-\frac{1}{c}\right)\Gamma^{3}\left(\frac{1}{c}+1\right)+\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(\frac{3}{c}+1\right)\right.\\
+ & & \left.-3\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\Gamma\left(\frac{2}{c}+1\right)\right]\\
+\gamma_{2} & = & -3+\frac{1}{k^{2}}\left[6\Gamma\left(1-\frac{2}{c}\right)\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\Gamma\left(\frac{2}{c}+1\right)\right.\\
+ & & -3\Gamma^{4}\left(1-\frac{1}{c}\right)\Gamma^{4}\left(\frac{1}{c}+1\right)+\Gamma\left(1-\frac{4}{c}\right)\Gamma\left(\frac{4}{c}+1\right)\\
+ & & \left.-4\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\Gamma\left(\frac{3}{c}+1\right)\right]\\
+m_{d} & = & \left(\frac{c-1}{c+1}\right)^{1/c}\,\textrm{if }c>1\,\textrm{otherwise }0\\
+m_{n} & = & 1\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=2-\log c.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Folded Cauchy
+\end_layout
+
+\begin_layout Standard
+This formula can be expressed in terms of the standard formulas for the
+ Cauchy distribution (call the cdf
+\begin_inset Formula $C\left(x\right)$
+\end_inset
+
+ and the pdf
+\begin_inset Formula $d\left(x\right)$
+\end_inset
+
+).
+ if
+\begin_inset Formula $Y$
+\end_inset
+
+ is cauchy then
+\begin_inset Formula $\left|Y\right|$
+\end_inset
+
+ is folded cauchy.
+ Note that
+\begin_inset Formula $x\geq0.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{1}{\pi\left(1+\left(x-c\right)^{2}\right)}+\frac{1}{\pi\left(1+\left(x+c\right)^{2}\right)}\\
+F\left(x;c\right) & = & \frac{1}{\pi}\tan^{-1}\left(x-c\right)+\frac{1}{\pi}\tan^{-1}\left(x+c\right)\\
+G\left(q;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+No moments
+\end_layout
+
+\begin_layout Section
+Folded Normal
+\end_layout
+
+\begin_layout Standard
+If
+\begin_inset Formula $Z$
+\end_inset
+
+ is Normal with mean
+\begin_inset Formula $L$
+\end_inset
+
+ and
+\begin_inset Formula $\sigma=S$
+\end_inset
+
+, then
+\begin_inset Formula $\left|Z\right|$
+\end_inset
+
+ is a folded normal with shape parameter
+\begin_inset Formula $c=\left|L\right|/S$
+\end_inset
+
+, location parameter
+\begin_inset Formula $0$
+\end_inset
+
+ and scale parameter
+\begin_inset Formula $S$
+\end_inset
+
+.
+ This is a special case of the non-central chi distribution with one-degree
+ of freedom and non-centrality parameter
+\begin_inset Formula $c^{2}.$
+\end_inset
+
+ Note that
+\begin_inset Formula $c\geq0$
+\end_inset
+
+.
+ The standard form of the folded normal is
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \sqrt{\frac{2}{\pi}}\cosh\left(cx\right)\exp\left(-\frac{x^{2}+c^{2}}{2}\right)\\
+F\left(x;c\right) & = & \Phi\left(x-c\right)-\Phi\left(-x-c\right)=\Phi\left(x-c\right)+\Phi\left(x+c\right)-1\\
+G\left(\alpha;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\exp\left[\frac{t}{2}\left(t-2c\right)\right]\left(1+e^{2ct}\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+k & = & \textrm{erf}\left(\frac{c}{\sqrt{2}}\right)\\
+p & = & \exp\left(-\frac{c^{2}}{2}\right)\\
+\mu & = & \sqrt{\frac{2}{\pi}}p+ck\\
+\mu_{2} & = & c^{2}+1-\mu^{2}\\
+\gamma_{1} & = & \frac{\sqrt{\frac{2}{\pi}}p^{3}\left(4-\frac{\pi}{p^{2}}\left(2c^{2}+1\right)\right)+2ck\left(6p^{2}+3cpk\sqrt{2\pi}+\pi c\left(k^{2}-1\right)\right)}{\pi\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{c^{4}+6c^{2}+3+6\left(c^{2}+1\right)\mu^{2}-3\mu^{4}-4p\mu\left(\sqrt{\frac{2}{\pi}}\left(c^{2}+2\right)+\frac{ck}{p}\left(c^{2}+3\right)\right)}{\mu_{2}^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Fratio (or F)
+\end_layout
+
+\begin_layout Standard
+Defined for
+\begin_inset Formula $x>0$
+\end_inset
+
+.
+ The distribution of
+\begin_inset Formula $\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)$
+\end_inset
+
+ if
+\begin_inset Formula $X_{1}$
+\end_inset
+
+ is chi-squared with
+\begin_inset Formula $v_{1}$
+\end_inset
+
+ degrees of freedom and
+\begin_inset Formula $X_{2}$
+\end_inset
+
+ is chi-squared with
+\begin_inset Formula $v_{2}$
+\end_inset
+
+ degrees of freedom.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\nu_{1},\nu_{2}\right) & = & \frac{\nu_{2}^{\nu_{2}/2}\nu_{1}^{\nu_{1}/2}x^{\nu_{1}/2-1}}{\left(\nu_{2}+\nu_{1}x\right)^{\left(\nu_{1}+\nu_{2}\right)/2}B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)}\\
+F\left(x;v_{1},v_{2}\right) & = & I\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2},\frac{\nu_{2}x}{\nu_{2}+\nu_{1}x}\right)\\
+G\left(q;\nu_{1},\nu_{2}\right) & = & \left[\frac{\nu_{2}}{I^{-1}\left(\nu_{1}/2,\nu_{2}/2,q\right)}-\frac{\nu_{1}}{\nu_{2}}\right]^{-1}.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{\nu_{2}}{\nu_{2}-2}\quad\nu_{2}>2\\
+\mu_{2} & = & \frac{2\nu_{2}^{2}\left(\nu_{1}+\nu_{2}-2\right)}{\nu_{1}\left(\nu_{2}-2\right)^{2}\left(\nu_{2}-4\right)}\quad v_{2}>4\\
+\gamma_{1} & = & \frac{2\left(2\nu_{1}+\nu_{2}-2\right)}{\nu_{2}-6}\sqrt{\frac{2\left(\nu_{2}-4\right)}{\nu_{1}\left(\nu_{1}+\nu_{2}-2\right)}}\quad\nu_{2}>6\\
+\gamma_{2} & = & \frac{3\left[8+\left(\nu_{2}-6\right)\gamma_{1}^{2}\right]}{2\nu-16}\quad\nu_{2}>8\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)
+\end_layout
+
+\begin_layout Standard
+A type of extreme-value distribution with a lower bound.
+ Defined for
+\begin_inset Formula $x>0$
+\end_inset
+
+ and
+\begin_inset Formula $c>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & cx^{c-1}\exp\left(-x^{c}\right)\\
+F\left(x;c\right) & = & 1-\exp\left(-x^{c}\right)\\
+G\left(q;c\right) & = & \left[-\log\left(1-q\right)\right]^{1/c}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \Gamma\left(1+\frac{1}{c}\right)\\
+\mu_{2} & = & \Gamma\left(1+\frac{2}{c}\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\\
+\gamma_{1} & = & \frac{\Gamma\left(1+\frac{3}{c}\right)-3\Gamma\left(1+\frac{2}{c}\right)\Gamma\left(1+\frac{1}{c}\right)+2\Gamma^{3}\left(1+\frac{1}{c}\right)}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\Gamma\left(1+\frac{4}{c}\right)-4\Gamma\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{3}{c}\right)+6\Gamma^{2}\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{2}{c}\right)-\Gamma^{4}\left(1+\frac{1}{c}\right)}{\mu_{2}^{2}}-3\\
+m_{d} & = & \left(\frac{c}{1+c}\right)^{1/c}\\
+m_{n} & = & G\left(\frac{1}{2};c\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
+
+\end_inset
+
+ where
+\begin_inset Formula $\gamma$
+\end_inset
+
+ is Euler's constant and equal to
+\begin_inset Formula \[
+\gamma\approx0.57721566490153286061.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)
+\end_layout
+
+\begin_layout Standard
+Defined for
+\begin_inset Formula $x<0$
+\end_inset
+
+ and
+\begin_inset Formula $c>0$
+\end_inset
+
+.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & c\left(-x\right)^{c-1}\exp\left(-\left(-x\right)^{c}\right)\\
+F\left(x;c\right) & = & \exp\left(-\left(-x\right)^{c}\right)\\
+G\left(q;c\right) & = & -\left(-\log q\right)^{1/c}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The mean is the negative of the right-skewed Frechet distribution given
+ above, and the other statistical parameters can be computed from
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
+
+\end_inset
+
+ where
+\begin_inset Formula $\gamma$
+\end_inset
+
+ is Euler's constant and equal to
+\begin_inset Formula \[
+\gamma\approx0.57721566490153286061.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gamma
+\end_layout
+
+\begin_layout Standard
+The standard form for the gamma distribution is
+\begin_inset Formula $\left(\alpha>0\right)$
+\end_inset
+
+ valid for
+\begin_inset Formula $x\geq0$
+\end_inset
+
+.
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\alpha\right) & = & \frac{1}{\Gamma\left(\alpha\right)}x^{\alpha-1}e^{-x}\\
+F\left(x;\alpha\right) & = & \Gamma\left(\alpha,x\right)\\
+G\left(q;\alpha\right) & = & \Gamma^{-1}\left(\alpha,q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \alpha\\
+\mu_{2} & = & \alpha\\
+\gamma_{1} & = & \frac{2}{\sqrt{\alpha}}\\
+\gamma_{2} & = & \frac{6}{\alpha}\\
+m_{d} & = & \alpha-1\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(a\right)\]
+
+\end_inset
+
+ where
+\begin_inset Formula \[
+\Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Generalized Logistic
+\end_layout
+
+\begin_layout Standard
+Has been used in the analysis of extreme values.
+ Has one shape parameter
+\begin_inset Formula $c>0.$
+\end_inset
+
+ And
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{c\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{c+1}}\\
+F\left(x;c\right) & = & \frac{1}{\left[1+\exp\left(-x\right)\right]^{c}}\\
+G\left(q;c\right) & = & -\log\left(q^{-1/c}-1\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+M\left(t\right)=\frac{c}{1-t}\,_{2}F_{1}\left(1+c,\,1-t\,;\,2-t\,;-1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \gamma+\psi_{0}\left(c\right)\\
+\mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\\
+\gamma_{1} & = & \frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\mu_{2}^{2}}\\
+m_{d} & = & \log c\\
+m_{n} & = & -\log\left(2^{1/c}-1\right)\end{eqnarray*}
+
+\end_inset
+
+ Note that the polygamma function is
+\begin_inset Formula \begin{eqnarray*}
+\psi_{n}\left(z\right) & = & \frac{d^{n+1}}{dz^{n+1}}\log\Gamma\left(z\right)\\
+ & = & \left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\
+ & = & \left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\end{eqnarray*}
+
+\end_inset
+
+ where
+\begin_inset Formula $\zeta\left(k,x\right)$
+\end_inset
+
+ is a generalization of the Riemann zeta function called the Hurwitz zeta
+ function Note that
+\begin_inset Formula $\zeta\left(n\right)\equiv\zeta\left(n,1\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Generalized Pareto
+\end_layout
+
+\begin_layout Standard
+Shape parameter
+\begin_inset Formula $c\neq0$
+\end_inset
+
+ and defined for
+\begin_inset Formula $x\geq0$
+\end_inset
+
+ for all
+\begin_inset Formula $c$
+\end_inset
+
+ and
+\begin_inset Formula $x<\frac{1}{\left|c\right|}$
+\end_inset
+
+ if
+\begin_inset Formula $c$
+\end_inset
+
+ is negative.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \left(1+cx\right)^{-1-\frac{1}{c}}\\
+F\left(x;c\right) & = & 1-\frac{1}{\left(1+cx\right)^{1/c}}\\
+G\left(q;c\right) & = & \frac{1}{c}\left[\left(\frac{1}{1-q}\right)^{c}-1\right]\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+M\left(t\right)=\left\{ \begin{array}{cc}
+\left(-\frac{t}{c}\right)^{\frac{1}{c}}e^{-\frac{t}{c}}\left[\Gamma\left(1-\frac{1}{c}\right)+\Gamma\left(-\frac{1}{c},-\frac{t}{c}\right)-\pi\csc\left(\frac{\pi}{c}\right)/\Gamma\left(\frac{1}{c}\right)\right] & c>0\\
+\left(\frac{\left|c\right|}{t}\right)^{1/\left|c\right|}\Gamma\left[\frac{1}{\left|c\right|},\frac{t}{\left|c\right|}\right] & c<0\end{array}\right.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\frac{\left(-1\right)^{n}}{c^{n}}\sum_{k=0}^{n}\left(\begin{array}{c}
+n\\
+k\end{array}\right)\frac{\left(-1\right)^{k}}{1-ck}\quad cn<1\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu_{1}^{\prime} & = & \frac{1}{1-c}\quad c<1\\
+\mu_{2}^{\prime} & = & \frac{2}{\left(1-2c\right)\left(1-c\right)}\quad c<\frac{1}{2}\\
+\mu_{3}^{\prime} & = & \frac{6}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)}\quad c<\frac{1}{3}\\
+\mu_{4}^{\prime} & = & \frac{24}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)\left(1-4c\right)}\quad c<\frac{1}{4}\end{eqnarray*}
+
+\end_inset
+
+ Thus,
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \mu_{1}^{\prime}\\
+\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
+\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=1+c\quad c>0\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Generalized Exponential
+\end_layout
+
+\begin_layout Standard
+Three positive shape parameters for
+\begin_inset Formula $x\geq0.$
+\end_inset
+
+ Note that
+\begin_inset Formula $a,b,$
+\end_inset
+
+ and
+\begin_inset Formula $c$
+\end_inset
+
+ are all
+\begin_inset Formula $>0.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,b,c\right) & = & \left(a+b\left(1-e^{-cx}\right)\right)\exp\left[ax-bx+\frac{b}{c}\left(1-e^{-cx}\right)\right]\\
+F\left(x;a,b,c\right) & = & 1-\exp\left[ax-bx+\frac{b}{c}\left(1-e^{-cx}\right)\right]\\
+G\left(q;a,b,c\right) & = & F^{-1}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Generalized Extreme Value
+\end_layout
+
+\begin_layout Standard
+Extreme value distributions with shape parameter
+\begin_inset Formula $c$
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+For
+\begin_inset Formula $c>0$
+\end_inset
+
+ defined on
+\begin_inset Formula $-\infty<x\leq1/c.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \exp\left[-\left(1-cx\right)^{1/c}\right]\left(1-cx\right)^{1/c-1}\\
+F\left(x;c\right) & = & \exp\left[-\left(1-cx\right)^{1/c}\right]\\
+G\left(q;c\right) & = & \frac{1}{c}\left[1-\left(-\log q\right)^{c}\right]\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\frac{1}{c^{n}}\sum_{k=0}^{n}\left(\begin{array}{c}
+n\\
+k\end{array}\right)\left(-1\right)^{k}\Gamma\left(ck+1\right)\quad cn>-1\]
+
+\end_inset
+
+ So,
+\begin_inset Formula \begin{eqnarray*}
+\mu_{1}^{\prime} & = & \frac{1}{c}\left(1-\Gamma\left(1+c\right)\right)\quad c>-1\\
+\mu_{2}^{\prime} & = & \frac{1}{c^{2}}\left(1-2\Gamma\left(1+c\right)+\Gamma\left(1+2c\right)\right)\quad c>-\frac{1}{2}\\
+\mu_{3}^{\prime} & = & \frac{1}{c^{3}}\left(1-3\Gamma\left(1+c\right)+3\Gamma\left(1+2c\right)-\Gamma\left(1+3c\right)\right)\quad c>-\frac{1}{3}\\
+\mu_{4}^{\prime} & = & \frac{1}{c^{4}}\left(1-4\Gamma\left(1+c\right)+6\Gamma\left(1+2c\right)-4\Gamma\left(1+3c\right)+\Gamma\left(1+4c\right)\right)\quad c>-\frac{1}{4}\end{eqnarray*}
+
+\end_inset
+
+ For
+\begin_inset Formula $c<0$
+\end_inset
+
+ defined on
+\begin_inset Formula $\frac{1}{c}\leq x<\infty.$
+\end_inset
+
+ For
+\begin_inset Formula $c=0$
+\end_inset
+
+ defined over all space
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;0\right) & = & \exp\left[-e^{-x}\right]e^{-x}\\
+F\left(x;0\right) & = & \exp\left[-e^{-x}\right]\\
+G\left(q;0\right) & = & -\log\left(-\log q\right)\end{eqnarray*}
+
+\end_inset
+
+ This is just the (left-skewed) Gumbel distribution for c=0.
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \gamma=-\psi_{0}\left(1\right)\\
+\mu_{2} & = & \frac{\pi^{2}}{6}\\
+\gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
+\gamma_{2} & = & \frac{12}{5}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Generalized Gamma
+\end_layout
+
+\begin_layout Standard
+A general probability form that reduces to many common distributions:
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\begin_inset Formula $a>0$
+\end_inset
+
+ and
+\begin_inset Formula $c\neq0.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,c\right) & = & \frac{\left|c\right|x^{ca-1}}{\Gamma\left(a\right)}\exp\left(-x^{c}\right)\\
+F\left(x;a,c\right) & = & \begin{array}{cc}
+\frac{\Gamma\left(a,x^{c}\right)}{\Gamma\left(a\right)} & c>0\\
+1-\frac{\Gamma\left(a,x^{c}\right)}{\Gamma\left(a\right)} & c<0\end{array}\\
+G\left(q;a,c\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{1/c}\quad c>0\\
+ & & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)\left(1-q\right)\right]\right\} ^{1/c}\quad c<0\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)}\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{\Gamma\left(a+\frac{1}{c}\right)}{\Gamma\left(a\right)}\\
+\mu_{2} & = & \frac{\Gamma\left(a+\frac{2}{c}\right)}{\Gamma\left(a\right)}-\mu^{2}\\
+\gamma_{1} & = & \frac{\Gamma\left(a+\frac{3}{c}\right)/\Gamma\left(a\right)-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\Gamma\left(a+\frac{4}{c}\right)/\Gamma\left(a\right)-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\\
+m_{d} & = & \left(\frac{ac-1}{c}\right)^{1/c}.\end{eqnarray*}
+
+\end_inset
+
+ Special cases are Weibull
+\begin_inset Formula $\left(a=1\right)$
+\end_inset
+
+, half-normal
+\begin_inset Formula $\left(a=1/2,c=2\right)$
+\end_inset
+
+ and ordinary gamma distributions
+\begin_inset Formula $c=1.$
+\end_inset
+
+ If
+\begin_inset Formula $c=-1$
+\end_inset
+
+ then it is the inverted gamma distribution.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\log\Gamma\left(a\right)-\log\left|c\right|.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Generalized Half-Logistic
+\end_layout
+
+\begin_layout Standard
+For
+\begin_inset Formula $x\in\left[0,1/c\right]$
+\end_inset
+
+ and
+\begin_inset Formula $c>0$
+\end_inset
+
+ we have
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{2\left(1-cx\right)^{\frac{1}{c}-1}}{\left(1+\left(1-cx\right)^{1/c}\right)^{2}}\\
+F\left(x;c\right) & = & \frac{1-\left(1-cx\right)^{1/c}}{1+\left(1-cx\right)^{1/c}}\\
+G\left(q;c\right) & = & \frac{1}{c}\left[1-\left(\frac{1-q}{1+q}\right)^{c}\right]\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & 2-\left(2c+1\right)\log2.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gilbrat
+\end_layout
+
+\begin_layout Standard
+Special case of the log-normal with
+\begin_inset Formula $\sigma=1$
+\end_inset
+
+ and
+\begin_inset Formula $S=1.0$
+\end_inset
+
+ (typically also
+\begin_inset Formula $L=0.0$
+\end_inset
+
+)
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\sigma\right) & = & \frac{1}{x\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\log x\right)^{2}\right]\\
+F\left(x;\sigma\right) & = & \Phi\left(\log x\right)=\frac{1}{2}\left[1+\textrm{erf}\left(\frac{\log x}{\sqrt{2}}\right)\right]\\
+G\left(q;\sigma\right) & = & \exp\left\{ \Phi^{-1}\left(q\right)\right\} \end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \sqrt{e}\\
+\mu_{2} & = & e\left[e-1\right]\\
+\gamma_{1} & = & \sqrt{e-1}\left(2+e\right)\\
+\gamma_{2} & = & e^{4}+2e^{3}+3e^{2}-6\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\
+ & \approx & 1.4189385332046727418\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gompertz (Truncated Gumbel)
+\end_layout
+
+\begin_layout Standard
+For
+\begin_inset Formula $x\geq0$
+\end_inset
+
+ and
+\begin_inset Formula $c>0$
+\end_inset
+
+.
+ In JKB the two shape parameters
+\begin_inset Formula $b,a$
+\end_inset
+
+ are reduced to the single shape-parameter
+\begin_inset Formula $c=b/a$
+\end_inset
+
+.
+ As
+\begin_inset Formula $a$
+\end_inset
+
+ is just a scale parameter when
+\begin_inset Formula $a\neq0$
+\end_inset
+
+.
+ If
+\begin_inset Formula $a=0,$
+\end_inset
+
+ the distribution reduces to the exponential distribution scaled by
+\begin_inset Formula $1/b.$
+\end_inset
+
+ Thus, the standard form is given as
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & ce^{x}\exp\left[-c\left(e^{x}-1\right)\right]\\
+F\left(x;c\right) & = & 1-\exp\left[-c\left(e^{x}-1\right)\right]\\
+G\left(q;c\right) & = & \log\left[1-\frac{1}{c}\log\left(1-q\right)\right]\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=1-\log\left(c\right)-e^{c}\textrm{Ei}\left(1,c\right),\]
+
+\end_inset
+
+where
+\begin_inset Formula \[
+\textrm{Ei}\left(n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\right)dt\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)
+\end_layout
+
+\begin_layout Standard
+One of a clase of extreme value distributions (right-skewed).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \exp\left(-\left(x+e^{-x}\right)\right)\\
+F\left(x\right) & = & \exp\left(-e^{-x}\right)\\
+G\left(q\right) & = & -\log\left(-\log\left(q\right)\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\Gamma\left(1-t\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \gamma=-\psi_{0}\left(1\right)\\
+\mu_{2} & = & \frac{\pi^{2}}{6}\\
+\gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
+\gamma_{2} & = & \frac{12}{5}\\
+m_{d} & = & 0\\
+m_{n} & = & -\log\left(\log2\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]\approx1.0608407169541684911\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gumbel Left-skewed (for minimum order statistic)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \exp\left(x-e^{x}\right)\\
+F\left(x\right) & = & 1-\exp\left(-e^{x}\right)\\
+G\left(q\right) & = & \log\left(-\log\left(1-q\right)\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\Gamma\left(1+t\right)\]
+
+\end_inset
+
+ Note, that
+\begin_inset Formula $\mu$
+\end_inset
+
+ is negative the mean for the right-skewed distribution.
+ Similar for median and mode.
+ All other moments are the same.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]\approx1.0608407169541684911.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+HalfCauchy
+\end_layout
+
+\begin_layout Standard
+If
+\begin_inset Formula $Z$
+\end_inset
+
+ is Hyperbolic Secant distributed then
+\begin_inset Formula $e^{Z}$
+\end_inset
+
+ is Half-Cauchy distributed.
+ Also, if
+\begin_inset Formula $W$
+\end_inset
+
+ is (standard) Cauchy distributed, then
+\begin_inset Formula $\left|W\right|$
+\end_inset
+
+ is Half-Cauchy distributed.
+ Special case of the Folded Cauchy distribution with
+\begin_inset Formula $c=0.$
+\end_inset
+
+ The standard form is
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{2}{\pi\left(1+x^{2}\right)}I_{[0,\infty)}\left(x\right)\\
+F\left(x\right) & = & \frac{2}{\pi}\arctan\left(x\right)I_{\left[0,\infty\right]}\left(x\right)\\
+G\left(q\right) & = & \tan\left(\frac{\pi}{2}q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\cos t+\frac{2}{\pi}\left[\textrm{Si}\left(t\right)\cos t-\textrm{Ci}\left(\textrm{-}t\right)\sin t\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+m_{d} & = & 0\\
+m_{n} & = & \tan\left(\frac{\pi}{4}\right)\end{eqnarray*}
+
+\end_inset
+
+ No moments, as the integrals diverge.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(2\pi\right)\\
+ & \approx & 1.8378770664093454836.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+HalfNormal
+\end_layout
+
+\begin_layout Standard
+This is a special case of the chi distribution with
+\begin_inset Formula $L=a$
+\end_inset
+
+ and
+\begin_inset Formula $S=b$
+\end_inset
+
+ and
+\begin_inset Formula $\nu=1.$
+\end_inset
+
+ This is also a special case of the folded normal with shape parameter
+\begin_inset Formula $c=0$
+\end_inset
+
+ and
+\begin_inset Formula $S=S.$
+\end_inset
+
+ If
+\begin_inset Formula $Z$
+\end_inset
+
+ is (standard) normally distributed then,
+\begin_inset Formula $\left|Z\right|$
+\end_inset
+
+ is half-normal.
+ The standard form is
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \sqrt{\frac{2}{\pi}}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\
+F\left(x\right) & = & 2\Phi\left(x\right)-1\\
+G\left(q\right) & = & \Phi^{-1}\left(\frac{1+q}{2}\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \sqrt{\frac{2}{\pi}}\\
+\mu_{2} & = & 1-\frac{2}{\pi}\\
+\gamma_{1} & = & \frac{\sqrt{2}\left(4-\pi\right)}{\left(\pi-2\right)^{3/2}}\\
+\gamma_{2} & = & \frac{8\left(\pi-3\right)}{\left(\pi-2\right)^{2}}\\
+m_{d} & = & 0\\
+m_{n} & = & \Phi^{-1}\left(\frac{3}{4}\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(\sqrt{\frac{\pi e}{2}}\right)\\
+ & \approx & 0.72579135264472743239.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Half-Logistic
+\end_layout
+
+\begin_layout Standard
+In the limit as
+\begin_inset Formula $c\rightarrow\infty$
+\end_inset
+
+ for the generalized half-logistic we have the half-logistic defined over
+
+\begin_inset Formula $x\geq0.$
+\end_inset
+
+ Also, the distribution of
+\begin_inset Formula $\left|X\right|$
+\end_inset
+
+ where
+\begin_inset Formula $X$
+\end_inset
+
+ has logistic distribtution.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{2e^{-x}}{\left(1+e^{-x}\right)^{2}}=\frac{1}{2}\textrm{sech}^{2}\left(\frac{x}{2}\right)\\
+F\left(x\right) & = & \frac{1-e^{-x}}{1+e^{-x}}=\tanh\left(\frac{x}{2}\right)\\
+G\left(q\right) & = & \log\left(\frac{1+q}{1-q}\right)=2\textrm{arctanh}\left(q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+M\left(t\right)=1-t\psi_{0}\left(\frac{1}{2}-\frac{t}{2}\right)+t\psi_{0}\left(1-\frac{t}{2}\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=2\left(1-2^{1-n}\right)n!\zeta\left(n\right)\quad n\neq1\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu_{1}^{\prime} & = & 2\log\left(2\right)\\
+\mu_{2}^{\prime} & = & 2\zeta\left(2\right)=\frac{\pi^{2}}{3}\\
+\mu_{3}^{\prime} & = & 9\zeta\left(3\right)\\
+\mu_{4}^{\prime} & = & 42\zeta\left(4\right)=\frac{7\pi^{4}}{15}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & 2-\log\left(2\right)\\
+ & \approx & 1.3068528194400546906.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Hyperbolic Secant
+\end_layout
+
+\begin_layout Standard
+Related to the logistic distribution and used in lifetime analysis.
+ Standard form is (defined over all
+\begin_inset Formula $x$
+\end_inset
+
+)
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{\pi}\textrm{sech}\left(x\right)\\
+F\left(x\right) & = & \frac{2}{\pi}\arctan\left(e^{x}\right)\\
+G\left(q\right) & = & \log\left(\tan\left(\frac{\pi}{2}q\right)\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\sec\left(\frac{\pi}{2}t\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu_{n}^{\prime} & = & \frac{1+\left(-1\right)^{n}}{2\pi2^{2n}}n!\left[\zeta\left(n+1,\frac{1}{4}\right)-\zeta\left(n+1,\frac{3}{4}\right)\right]\\
+ & = & \left\{ \begin{array}{cc}
+0 & n\textrm{ odd}\\
+C_{n/2}\frac{\pi^{n}}{2^{n}} & n\textrm{ even}\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+ where
+\begin_inset Formula $C_{m}$
+\end_inset
+
+ is an integer given by
+\begin_inset Formula \begin{eqnarray*}
+C_{m} & = & \frac{\left(2m\right)!\left[\zeta\left(2m+1,\frac{1}{4}\right)-\zeta\left(2m+1,\frac{3}{4}\right)\right]}{\pi^{2m+1}2^{2m}}\\
+ & = & 4\left(-1\right)^{m-1}\frac{16^{m}}{2m+1}B_{2m+1}\left(\frac{1}{4}\right)\end{eqnarray*}
+
+\end_inset
+
+where
+\begin_inset Formula $B_{2m+1}\left(\frac{1}{4}\right)$
+\end_inset
+
+ is the Bernoulli polynomial of order
+\begin_inset Formula $2m+1$
+\end_inset
+
+ evaluated at
+\begin_inset Formula $1/4.$
+\end_inset
+
+ Thus
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\left\{ \begin{array}{cc}
+0 & n\textrm{ odd}\\
+4\left(-1\right)^{n/2-1}\frac{\left(2\pi\right)^{n}}{n+1}B_{n+1}\left(\frac{1}{4}\right) & n\textrm{ even}\end{array}\right.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+m_{d}=m_{n}=\mu & = & 0\\
+\mu_{2} & = & \frac{\pi^{2}}{4}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & 2\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=\log\left(2\pi\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gauss Hypergeometric
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $x\in\left[0,1\right]$
+\end_inset
+
+,
+\begin_inset Formula $\alpha>0,\,\beta>0$
+\end_inset
+
+
+\begin_inset Formula \[
+C^{-1}=B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\alpha,\beta,\gamma,z\right) & = & Cx^{\alpha-1}\frac{\left(1-x\right)^{\beta-1}}{\left(1+zx\right)^{\gamma}}\\
+\mu_{n}^{\prime} & = & \frac{B\left(n+\alpha,\beta\right)}{B\left(\alpha,\beta\right)}\frac{\,_{2}F_{1}\left(\gamma,\alpha+n;\alpha+\beta+n;-z\right)}{\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Inverted Gamma
+\end_layout
+
+\begin_layout Standard
+Special case of the generalized Gamma distribution with
+\begin_inset Formula $c=-1$
+\end_inset
+
+ and
+\begin_inset Formula $a>0$
+\end_inset
+
+,
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a\right) & = & \frac{x^{-a-1}}{\Gamma\left(a\right)}\exp\left(-\frac{1}{x}\right)\\
+F\left(x;a\right) & = & \frac{\Gamma\left(a,\frac{1}{x}\right)}{\Gamma\left(a\right)}\\
+G\left(q;a\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{-1}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{1}{a-1}\quad a>1\\
+\mu_{2} & = & \frac{1}{\left(a-2\right)\left(a-1\right)}-\mu^{2}\quad a>2\\
+\gamma_{1} & = & \frac{\frac{1}{\left(a-3\right)\left(a-2\right)\left(a-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\frac{1}{\left(a-4\right)\left(a-3\right)\left(a-2\right)\left(a-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+m_{d}=\frac{1}{a+1}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=a-\left(a+1\right)\Psi\left(a\right)+\log\Gamma\left(a\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Inverse Normal (Inverse Gaussian)
+\end_layout
+
+\begin_layout Standard
+The standard form involves the shape parameter
+\begin_inset Formula $\mu$
+\end_inset
+
+ (in most definitions,
+\begin_inset Formula $L=0.0$
+\end_inset
+
+ is used).
+ (In terms of the regress documentation
+\begin_inset Formula $\mu=A/B$
+\end_inset
+
+) and
+\begin_inset Formula $B=S$
+\end_inset
+
+ and
+\begin_inset Formula $L$
+\end_inset
+
+ is not a parameter in that distribution.
+ A standard form is
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2x\mu^{2}}\right).\\
+F\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\\
+G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \mu\\
+\mu_{2} & = & \mu^{3}\\
+\gamma_{1} & = & 3\sqrt{\mu}\\
+\gamma_{2} & = & 15\mu\\
+m_{d} & = & \frac{\mu}{2}\left(\sqrt{9\mu^{2}+4}-3\mu\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This is related to the canonical form or JKB
+\begin_inset Quotes eld
+\end_inset
+
+two-parameter
+\begin_inset Quotes erd
+\end_inset
+
+ inverse Gaussian when written in it's full form with scale parameter
+\begin_inset Formula $S$
+\end_inset
+
+ and location parameter
+\begin_inset Formula $L$
+\end_inset
+
+ by taking
+\begin_inset Formula $L=0$
+\end_inset
+
+ and
+\begin_inset Formula $S\equiv\lambda,$
+\end_inset
+
+ then
+\begin_inset Formula $\mu S$
+\end_inset
+
+ is equal to
+\begin_inset Formula $\mu_{2}$
+\end_inset
+
+ where
+\begin_inset Formula $\mu_{2}$
+\end_inset
+
+ is the parameter used by JKB.
+ We prefer this form because of it's consistent use of the scale parameter.
+ Notice that in JKB the skew
+\begin_inset Formula $\left(\sqrt{\beta_{1}}\right)$
+\end_inset
+
+ and the kurtosis (
+\begin_inset Formula $\beta_{2}-3$
+\end_inset
+
+) are both functions only of
+\begin_inset Formula $\mu_{2}/\lambda=\mu S/S=\mu$
+\end_inset
+
+ as shown here, while the variance and mean of the standard form here are
+ transformed appropriately.
+
+\end_layout
+
+\begin_layout Section
+Inverted Weibull
+\end_layout
+
+\begin_layout Standard
+Shape parameter
+\begin_inset Formula $c>0$
+\end_inset
+
+ and
+\begin_inset Formula $x>0$
+\end_inset
+
+.
+ Then
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & cx^{-c-1}\exp\left(-x^{-c}\right)\\
+F\left(x;c\right) & = & \exp\left(-x^{-c}\right)\\
+G\left(q;c\right) & = & \left(-\log q\right)^{-1/c}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=1+\gamma+\frac{\gamma}{c}-\log\left(c\right)\]
+
+\end_inset
+
+ where
+\begin_inset Formula $\gamma$
+\end_inset
+
+ is Euler's constant.
+\end_layout
+
+\begin_layout Section
+Johnson SB
+\end_layout
+
+\begin_layout Standard
+Defined for
+\begin_inset Formula $x\in\left(0,1\right)$
+\end_inset
+
+ with two shape parameters
+\begin_inset Formula $a$
+\end_inset
+
+ and
+\begin_inset Formula $b>0.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,b\right) & = & \frac{b}{x\left(1-x\right)}\phi\left(a+b\log\frac{x}{1-x}\right)\\
+F\left(x;a,b\right) & = & \Phi\left(a+b\log\frac{x}{1-x}\right)\\
+G\left(q;a,b\right) & = & \frac{1}{1+\exp\left[-\frac{1}{b}\left(\Phi^{-1}\left(q\right)-a\right)\right]}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Johnson SU
+\end_layout
+
+\begin_layout Standard
+Defined for all
+\begin_inset Formula $x$
+\end_inset
+
+ with two shape parameters
+\begin_inset Formula $a$
+\end_inset
+
+ and
+\begin_inset Formula $b>0$
+\end_inset
+
+.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,b\right) & = & \frac{b}{\sqrt{x^{2}+1}}\phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\
+F\left(x;a,b\right) & = & \Phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\
+G\left(q;a,b\right) & = & \sinh\left[\frac{\Phi^{-1}\left(q\right)-a}{b}\right]\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+KSone
+\end_layout
+
+\begin_layout Section
+KStwo
+\end_layout
+
+\begin_layout Section
+Laplace (Double Exponential, Bilateral Expoooonential)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{2}e^{-\left|x\right|}\\
+F\left(x\right) & = & \left\{ \begin{array}{ccc}
+\frac{1}{2}e^{x} & & x\leq0\\
+1-\frac{1}{2}e^{-x} & & x>0\end{array}\right.\\
+G\left(q\right) & = & \left\{ \begin{array}{ccc}
+\log\left(2q\right) & & q\leq\frac{1}{2}\\
+-\log\left(2-2q\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+m_{d}=m_{n}=\mu & = & 0\\
+\mu_{2} & = & 2\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & 3\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The ML estimator of the location parameter is
+\begin_inset Formula \[
+\hat{L}=\textrm{median}\left(X_{i}\right)\]
+
+\end_inset
+
+ where
+\begin_inset Formula $X_{i}$
+\end_inset
+
+ is a sequence of
+\begin_inset Formula $N$
+\end_inset
+
+ mutually independent Laplace RV's and the median is some number between
+ the
+\begin_inset Formula $\frac{1}{2}N\textrm{th}$
+\end_inset
+
+ and the
+\begin_inset Formula $(N/2+1)\textrm{th}$
+\end_inset
+
+ order statistic (
+\emph on
+e.g.
+
+\emph default
+ take the average of these two) when
+\begin_inset Formula $N$
+\end_inset
+
+ is even.
+ Also,
+\begin_inset Formula \[
+\hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|.\]
+
+\end_inset
+
+ Replace
+\begin_inset Formula $\hat{L}$
+\end_inset
+
+ with
+\begin_inset Formula $L$
+\end_inset
+
+ if it is known.
+ If
+\begin_inset Formula $L$
+\end_inset
+
+ is known then this estimator is distributed as
+\begin_inset Formula $\left(2N\right)^{-1}S\cdot\chi_{2N}^{2}$
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(2e\right)\\
+ & \approx & 1.6931471805599453094.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Left-skewed Lévy
+\end_layout
+
+\begin_layout Standard
+Special case of Lévy-stable distribution with
+\begin_inset Formula $\alpha=\frac{1}{2}$
+\end_inset
+
+ and
+\begin_inset Formula $\beta=-1$
+\end_inset
+
+ the support is
+\begin_inset Formula $x<0$
+\end_inset
+
+.
+ In standard form
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{\left|x\right|\sqrt{2\pi\left|x\right|}}\exp\left(-\frac{1}{2\left|x\right|}\right)\\
+F\left(x\right) & = & 2\Phi\left(\frac{1}{\sqrt{\left|x\right|}}\right)-1\\
+G\left(q\right) & = & -\left[\Phi^{-1}\left(\frac{q+1}{2}\right)\right]^{-2}.\end{eqnarray*}
+
+\end_inset
+
+No moments.
+\end_layout
+
+\begin_layout Section
+Lévy
+\end_layout
+
+\begin_layout Standard
+A special case of Lévy-stable distributions with
+\begin_inset Formula $\alpha=\frac{1}{2}$
+\end_inset
+
+ and
+\begin_inset Formula $\beta=1$
+\end_inset
+
+.
+ In standard form it is defined for
+\begin_inset Formula $x>0$
+\end_inset
+
+ as
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{x\sqrt{2\pi x}}\exp\left(-\frac{1}{2x}\right)\\
+F\left(x\right) & = & 2\left[1-\Phi\left(\frac{1}{\sqrt{x}}\right)\right]\\
+G\left(q\right) & = & \left[\Phi^{-1}\left(1-\frac{q}{2}\right)\right]^{-2}.\end{eqnarray*}
+
+\end_inset
+
+ It has no finite moments.
+\end_layout
+
+\begin_layout Section
+Logistic (Sech-squared)
+\end_layout
+
+\begin_layout Standard
+A special case of the Generalized Logistic distribution with
+\begin_inset Formula $c=1.$
+\end_inset
+
+ Defined for
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{2}}\\
+F\left(x\right) & = & \frac{1}{1+\exp\left(-x\right)}\\
+G\left(q\right) & = & -\log\left(1/q-1\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \gamma+\psi_{0}\left(1\right)=0\\
+\mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(1\right)=\frac{\pi^{2}}{3}\\
+\gamma_{1} & = & \frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}=0\\
+\gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\mu_{2}^{2}}=\frac{6}{5}\\
+m_{d} & = & \log1=0\\
+m_{n} & = & -\log\left(2-1\right)=0\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=1.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Log Double Exponential (Log-Laplace)
+\end_layout
+
+\begin_layout Standard
+Defined over
+\begin_inset Formula $x>0$
+\end_inset
+
+ with
+\begin_inset Formula $c>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \left\{ \begin{array}{ccc}
+\frac{c}{2}x^{c-1} & & 0<x<1\\
+\frac{c}{2}x^{-c-1} & & x\geq1\end{array}\right.\\
+F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
+\frac{1}{2}x^{c} & & 0<x<1\\
+1-\frac{1}{2}x^{-c} & & x\geq1\end{array}\right.\\
+G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
+\left(2q\right)^{1/c} & & 0\leq q<\frac{1}{2}\\
+\left(2-2q\right)^{-1/c} & & \frac{1}{2}\leq q\leq1\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=\log\left(\frac{2e}{c}\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Log Gamma
+\end_layout
+
+\begin_layout Standard
+A single shape parameter
+\begin_inset Formula $c>0$
+\end_inset
+
+ (Defined for all
+\begin_inset Formula $x$
+\end_inset
+
+)
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{\exp\left(cx-e^{x}\right)}{\Gamma\left(c\right)}\\
+F\left(x;c\right) & = & \frac{\Gamma\left(c,e^{x}\right)}{\Gamma\left(c\right)}\\
+G\left(q;c\right) & = & \log\left[\Gamma^{-1}\left[c,q\Gamma\left(c\right)\right]\right]\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\int_{0}^{\infty}\left[\log y\right]^{n}y^{c-1}\exp\left(-y\right)dy.\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \mu_{1}^{\prime}\\
+\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
+\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Log Normal (Cobb-Douglass)
+\end_layout
+
+\begin_layout Standard
+Has one shape parameter
+\begin_inset Formula $\sigma$
+\end_inset
+
+>0.
+ (Notice that the
+\begin_inset Quotes eld
+\end_inset
+
+Regress
+\begin_inset Quotes erd
+\end_inset
+
+
+\begin_inset Formula $A=\log S$
+\end_inset
+
+ where
+\begin_inset Formula $S$
+\end_inset
+
+ is the scale parameter and
+\begin_inset Formula $A$
+\end_inset
+
+ is the mean of the underlying normal distribution).
+ The standard form is
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\sigma\right) & = & \frac{1}{\sigma x\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{\log x}{\sigma}\right)^{2}\right]\\
+F\left(x;\sigma\right) & = & \Phi\left(\frac{\log x}{\sigma}\right)\\
+G\left(q;\sigma\right) & = & \exp\left\{ \sigma\Phi^{-1}\left(q\right)\right\} \end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \exp\left(\sigma^{2}/2\right)\\
+\mu_{2} & = & \exp\left(\sigma^{2}\right)\left[\exp\left(\sigma^{2}\right)-1\right]\\
+\gamma_{1} & = & \sqrt{p-1}\left(2+p\right)\\
+\gamma_{2} & = & p^{4}+2p^{3}+3p^{2}-6\quad\quad p=e^{\sigma^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Notice that using JKB notation we have
+\begin_inset Formula $\theta=L,$
+\end_inset
+
+
+\begin_inset Formula $\zeta=\log S$
+\end_inset
+
+ and we have given the so-called antilognormal form of the distribution.
+ This is more consistent with the location, scale parameter description
+ of general probability distributions.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right].\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Also, note that if
+\begin_inset Formula $X$
+\end_inset
+
+ is a log-normally distributed random-variable with
+\begin_inset Formula $L=0$
+\end_inset
+
+ and
+\begin_inset Formula $S$
+\end_inset
+
+ and shape parameter
+\begin_inset Formula $\sigma.$
+\end_inset
+
+ Then,
+\begin_inset Formula $\log X$
+\end_inset
+
+ is normally distributed with variance
+\begin_inset Formula $\sigma^{2}$
+\end_inset
+
+ and mean
+\begin_inset Formula $\log S.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Nakagami
+\end_layout
+
+\begin_layout Standard
+Generalization of the chi distribution.
+ Shape parameter is
+\begin_inset Formula $\nu>0.$
+\end_inset
+
+ Defined for
+\begin_inset Formula $x>0.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\nu\right) & = & \frac{2\nu^{\nu}}{\Gamma\left(\nu\right)}x^{2\nu-1}\exp\left(-\nu x^{2}\right)\\
+F\left(x;\nu\right) & = & \Gamma\left(\nu,\nu x^{2}\right)\\
+G\left(q;\nu\right) & = & \sqrt{\frac{1}{\nu}\Gamma^{-1}\left(v,q\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{\Gamma\left(\nu+\frac{1}{2}\right)}{\sqrt{\nu}\Gamma\left(\nu\right)}\\
+\mu_{2} & = & \left[1-\mu^{2}\right]\\
+\gamma_{1} & = & \frac{\mu\left(1-4v\mu_{2}\right)}{2\nu\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{-6\mu^{4}\nu+\left(8\nu-2\right)\mu^{2}-2\nu+1}{\nu\mu_{2}^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Noncentral beta*
+\end_layout
+
+\begin_layout Standard
+Defined over
+\begin_inset Formula $x\in\left[0,1\right]$
+\end_inset
+
+ with
+\begin_inset Formula $a>0$
+\end_inset
+
+ and
+\begin_inset Formula $b>0$
+\end_inset
+
+ and
+\begin_inset Formula $c\geq0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+F\left(x;a,b,c\right)=\sum_{j=0}^{\infty}\frac{e^{-c/2}\left(\frac{c}{2}\right)^{j}}{j!}I_{B}\left(a+j,b;0\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Noncentral chi*
+\end_layout
+
+\begin_layout Section
+Noncentral chi-squared
+\end_layout
+
+\begin_layout Standard
+The distribution of
+\begin_inset Formula $\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}$
+\end_inset
+
+ where
+\begin_inset Formula $Z_{i}$
+\end_inset
+
+ are independent standard normal variables and
+\begin_inset Formula $\delta_{i}$
+\end_inset
+
+ are constants.
+
+\begin_inset Formula $\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.$
+\end_inset
+
+ (In communications it is called the Marcum-Q function).
+ Can be thought of as a Generalized Rayleigh-Rice distribution.
+ For
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\nu,\lambda\right) & = & e^{-\left(\lambda+x\right)/2}\frac{1}{2}\left(\frac{x}{\lambda}\right)^{\left(\nu-2\right)/4}I_{\left(\nu-2\right)/2}\left(\sqrt{\lambda x}\right)\\
+F\left(x;\nu,\lambda\right) & = & \sum_{j=0}^{\infty}\left\{ \frac{\left(\lambda/2\right)^{j}}{j!}e^{-\lambda/2}\right\} \textrm{Pr}\left[\chi_{\nu+2j}^{2}\leq x\right]\\
+G\left(q;\nu,\lambda\right) & = & F^{-1}\left(x;\nu,\lambda\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \nu+\lambda\\
+\mu_{2} & = & 2\left(\nu+2\lambda\right)\\
+\gamma_{1} & = & \frac{\sqrt{8}\left(\nu+3\lambda\right)}{\left(\nu+2\lambda\right)^{3/2}}\\
+\gamma_{2} & = & \frac{12\left(\nu+4\lambda\right)}{\left(\nu+2\lambda\right)^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Noncentral F
+\end_layout
+
+\begin_layout Standard
+Let
+\begin_inset Formula $\lambda>0$
+\end_inset
+
+ and
+\begin_inset Formula $\nu_{1}>0$
+\end_inset
+
+ and
+\begin_inset Formula $\nu_{2}>0.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\lambda,\nu_{1},\nu_{2}\right) & = & \exp\left[\frac{\lambda}{2}+\frac{\left(\lambda\nu_{1}x\right)}{2\left(\nu_{1}x+\nu_{2}\right)}\right]\nu_{1}^{\nu_{1}/2}\nu_{2}^{\nu_{2}/2}x^{\nu_{1}/2-1}\\
+ & & \times\left(\nu_{2}+\nu_{1}x\right)^{-\left(\nu_{1}+\nu_{2}\right)/2}\frac{\Gamma\left(\frac{\nu_{1}}{2}\right)\Gamma\left(1+\frac{\nu_{2}}{2}\right)L_{\nu_{2}/2}^{\nu_{1}/2-1}\left(-\frac{\lambda\nu_{1}x}{2\left(\nu_{1}x+\nu_{2}\right)}\right)}{B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)\Gamma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Noncentral t
+\end_layout
+
+\begin_layout Standard
+The distribution of the ratio
+\begin_inset Formula \[
+\frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}}\]
+
+\end_inset
+
+ where
+\begin_inset Formula $U$
+\end_inset
+
+ and
+\begin_inset Formula $\chi_{\nu}$
+\end_inset
+
+ are independent and distributed as a standard normal and chi with
+\begin_inset Formula $\nu$
+\end_inset
+
+ degrees of freedom.
+ Note
+\begin_inset Formula $\lambda>0$
+\end_inset
+
+ and
+\begin_inset Formula $\nu>0$
+\end_inset
+
+.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\lambda,\nu\right) & = & \frac{\nu^{\nu/2}\Gamma\left(\nu+1\right)}{2^{\nu}e^{\lambda^{2}/2}\left(\nu+x^{2}\right)^{\nu/2}\Gamma\left(\nu/2\right)}\\
+ & & \times\left\{ \frac{\sqrt{2}\lambda x\,_{1}F_{1}\left(\frac{\nu}{2}+1;\frac{3}{2};\frac{\lambda^{2}x^{2}}{2\left(\nu+x^{2}\right)}\right)}{\left(\nu+x^{2}\right)\Gamma\left(\frac{\nu+1}{2}\right)}\right.\\
+ & & -\left.\frac{\,_{1}F_{1}\left(\frac{\nu+1}{2};\frac{1}{2};\frac{\lambda^{2}x^{2}}{2\left(\nu+x^{2}\right)}\right)}{\sqrt{\nu+x^{2}}\Gamma\left(\frac{\nu}{2}+1\right)}\right\} \\
+ & = & \frac{\Gamma\left(\nu+1\right)}{2^{\left(\nu-1\right)/2}\sqrt{\pi\nu}\Gamma\left(\nu/2\right)}\exp\left[-\frac{\nu\lambda^{2}}{\nu+x^{2}}\right]\\
+ & & \times\left(\frac{\nu}{\nu+x^{2}}\right)^{\left(\nu-1\right)/2}Hh_{\nu}\left(-\frac{\lambda x}{\sqrt{\nu+x^{2}}}\right)\\
+F\left(x;\lambda,\nu\right) & =\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Normal
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}\\
+F\left(x\right) & = & \Phi\left(x\right)=\frac{1}{2}+\frac{1}{2}\textrm{erf}\left(\frac{\textrm{x}}{\sqrt{2}}\right)\\
+G\left(q\right) & = & \Phi^{-1}\left(q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\align center
+\begin_inset Formula \begin{eqnarray*}
+m_{d}=m_{n}=\mu & = & 0\\
+\mu_{2} & = & 1\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & 0\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\
+ & \approx & 1.4189385332046727418\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Maxwell
+\end_layout
+
+\begin_layout Standard
+This is a special case of the Chi distribution with
+\begin_inset Formula $L=0$
+\end_inset
+
+ and
+\begin_inset Formula $S=S=\frac{1}{\sqrt{a}}$
+\end_inset
+
+ and
+\begin_inset Formula $\nu=3.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \sqrt{\frac{2}{\pi}}x^{2}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\
+F\left(x\right) & = & \Gamma\left(\frac{3}{2},\frac{x^{2}}{2}\right)\\
+G\left(\alpha\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\alpha\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 2\sqrt{\frac{2}{\pi}}\\
+\mu_{2} & = & 3-\frac{8}{\pi}\\
+\gamma_{1} & = & \sqrt{2}\frac{32-10\pi}{\left(3\pi-8\right)^{3/2}}\\
+\gamma_{2} & = & \frac{-12\pi^{2}+160\pi-384}{\left(3\pi-8\right)^{2}}\\
+m_{d} & = & \sqrt{2}\\
+m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\frac{1}{2}\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=\log\left(\sqrt{\frac{2\pi}{e}}\right)+\gamma.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Mielke's Beta-Kappa
+\end_layout
+
+\begin_layout Standard
+A generalized F distribution.
+ Two shape parameters
+\begin_inset Formula $\kappa$
+\end_inset
+
+ and
+\begin_inset Formula $\theta$
+\end_inset
+
+, and
+\begin_inset Formula $x>0$
+\end_inset
+
+.
+ The
+\begin_inset Formula $\beta$
+\end_inset
+
+ in the DATAPLOT reference is a scale parameter.
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\kappa,\theta\right) & = & \frac{\kappa x^{\kappa-1}}{\left(1+x^{\theta}\right)^{1+\frac{\kappa}{\theta}}}\\
+F\left(x;\kappa,\theta\right) & = & \frac{x^{\kappa}}{\left(1+x^{\theta}\right)^{\kappa/\theta}}\\
+G\left(q;\kappa,\theta\right) & = & \left(\frac{q^{\theta/\kappa}}{1-q^{\theta/\kappa}}\right)^{1/\theta}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Pareto
+\end_layout
+
+\begin_layout Standard
+For
+\begin_inset Formula $x\geq1$
+\end_inset
+
+ and
+\begin_inset Formula $b>0$
+\end_inset
+
+.
+ Standard form is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;b\right) & = & \frac{b}{x^{b+1}}\\
+F\left(x;b\right) & = & 1-\frac{1}{x^{b}}\\
+G\left(q;b\right) & = & \left(1-q\right)^{-1/b}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{b}{b-1}\quad b>1\\
+\mu_{2} & = & \frac{b}{\left(b-2\right)\left(b-1\right)^{2}}\quad b>2\\
+\gamma_{1} & = & \frac{2\left(b+1\right)\sqrt{b-2}}{\left(b-3\right)\sqrt{b}}\quad b>3\\
+\gamma_{2} & = & \frac{6\left(b^{3}+b^{2}-6b-2\right)}{b\left(b^{2}-7b+12\right)}\quad b>4\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left(X\right)=\frac{1}{c}+1-\log\left(c\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Pareto Second Kind (Lomax)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $c>0.$
+\end_inset
+
+ This is Pareto of the first kind with
+\begin_inset Formula $L=-1.0$
+\end_inset
+
+ so
+\begin_inset Formula $x\geq0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{c}{\left(1+x\right)^{c+1}}\\
+F\left(x;c\right) & = & 1-\frac{1}{\left(1+x\right)^{c}}\\
+G\left(q;c\right) & = & \left(1-q\right)^{-1/c}-1\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=\frac{1}{c}+1-\log\left(c\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Power Log Normal
+\end_layout
+
+\begin_layout Standard
+A generalization of the log-normal distribution
+\begin_inset Formula $\sigma>0$
+\end_inset
+
+ and
+\begin_inset Formula $c>0$
+\end_inset
+
+ and
+\begin_inset Formula $x>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\sigma,c\right) & = & \frac{c}{x\sigma}\phi\left(\frac{\log x}{\sigma}\right)\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c-1}\\
+F\left(x;\sigma,c\right) & = & 1-\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c}\\
+G\left(q;\sigma,c\right) & = & \exp\left[-\sigma\Phi^{-1}\left[\left(1-q\right)^{1/c}\right]\right]\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\int_{0}^{1}\exp\left[-n\sigma\Phi^{-1}\left(y^{1/c}\right)\right]dy\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \mu_{1}^{\prime}\\
+\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
+\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
+
+\end_inset
+
+ This distribution reduces to the log-normal distribution when
+\begin_inset Formula $c=1.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Power Normal
+\end_layout
+
+\begin_layout Standard
+A generalization of the normal distribution,
+\begin_inset Formula $c>0$
+\end_inset
+
+ for
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & c\phi\left(x\right)\left(\Phi\left(-x\right)\right)^{c-1}\\
+F\left(x;c\right) & = & 1-\left(\Phi\left(-x\right)\right)^{c}\\
+G\left(q;c\right) & = & -\Phi^{-1}\left[\left(1-q\right)^{1/c}\right]\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\left(-1\right)^{n}\int_{0}^{1}\left[\Phi^{-1}\left(y^{1/c}\right)\right]^{n}dy\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \mu_{1}^{\prime}\\
+\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
+\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
+\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
+
+\end_inset
+
+For
+\begin_inset Formula $c=1$
+\end_inset
+
+ this reduces to the normal distribution.
+
+\end_layout
+
+\begin_layout Section
+Power-function
+\end_layout
+
+\begin_layout Standard
+A special case of the beta distribution with
+\begin_inset Formula $b=1$
+\end_inset
+
+: defined for
+\begin_inset Formula $x\in\left[0,1\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+a>0\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a\right) & = & ax^{a-1}\\
+F\left(x;a\right) & = & x^{a}\\
+G\left(q;a\right) & = & q^{1/a}\\
+\mu & = & \frac{a}{a+1}\\
+\mu_{2} & = & \frac{a\left(a+2\right)}{\left(a+1\right)^{2}}\\
+\gamma_{1} & = & 2\left(1-a\right)\sqrt{\frac{a+2}{a\left(a+3\right)}}\\
+\gamma_{2} & = & \frac{6\left(a^{3}-a^{2}-6a+2\right)}{a\left(a+3\right)\left(a+4\right)}\\
+m_{d} & = & 1\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=1-\frac{1}{a}-\log\left(a\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+R-distribution
+\end_layout
+
+\begin_layout Standard
+A general-purpose distribution with a variety of shapes controlled by
+\begin_inset Formula $c>0.$
+\end_inset
+
+ Range of standard distribution is
+\begin_inset Formula $x\in\left[-1,1\right]$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{\left(1-x^{2}\right)^{c/2-1}}{B\left(\frac{1}{2},\frac{c}{2}\right)}\\
+F\left(x;c\right) & = & \frac{1}{2}+\frac{x}{B\left(\frac{1}{2},\frac{c}{2}\right)}\,_{2}F_{1}\left(\frac{1}{2},1-\frac{c}{2};\frac{3}{2};x^{2}\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\frac{\left(1+\left(-1\right)^{n}\right)}{2}B\left(\frac{n+1}{2},\frac{c}{2}\right)\]
+
+\end_inset
+
+ The R-distribution with parameter
+\begin_inset Formula $n$
+\end_inset
+
+ is the distribution of the correlation coefficient of a random sample of
+ size
+\begin_inset Formula $n$
+\end_inset
+
+ drawn from a bivariate normal distribution with
+\begin_inset Formula $\rho=0.$
+\end_inset
+
+ The mean of the standard distribution is always zero and as the sample
+ size grows, the distribution's mass concentrates more closely about this
+ mean.
+
+\end_layout
+
+\begin_layout Section
+Rayleigh
+\end_layout
+
+\begin_layout Standard
+This is Chi distribution with
+\begin_inset Formula $L=0.0$
+\end_inset
+
+ and
+\begin_inset Formula $\nu=2$
+\end_inset
+
+ and
+\begin_inset Formula $S=S$
+\end_inset
+
+ (no location parameter is generally used), the mode of the distribution
+ is
+\begin_inset Formula $S.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(r\right) & = & re^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\
+F\left(r\right) & = & 1-e^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\
+G\left(q\right) & = & \sqrt{-2\log\left(1-q\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \sqrt{\frac{\pi}{2}}\\
+\mu_{2} & = & \frac{4-\pi}{2}\\
+\gamma_{1} & = & \frac{2\left(\pi-3\right)\sqrt{\pi}}{\left(4-\pi\right)^{3/2}}\\
+\gamma_{2} & = & \frac{24\pi-6\pi^{2}-16}{\left(4-\pi\right)^{2}}\\
+m_{d} & = & 1\\
+m_{n} & = & \sqrt{2\log\left(2\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Rice*
+\end_layout
+
+\begin_layout Standard
+Defined for
+\begin_inset Formula $x>0$
+\end_inset
+
+ and
+\begin_inset Formula $b>0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;b\right) & = & x\exp\left(-\frac{x^{2}+b^{2}}{2}\right)I_{0}\left(xb\right)\\
+F\left(x;b\right) & = & \int_{0}^{x}\alpha\exp\left(-\frac{\alpha^{2}+b^{2}}{2}\right)I_{0}\left(\alpha b\right)d\alpha\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(1+\frac{n}{2}\right)\,_{1}F_{1}\left(-\frac{n}{2};1;-\frac{b^{2}}{2}\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Reciprocal
+\end_layout
+
+\begin_layout Standard
+Shape parameters
+\begin_inset Formula $a,b>0$
+\end_inset
+
+
+\begin_inset Formula $x\in\left[a,b\right]$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;a,b\right) & = & \frac{1}{x\log\left(b/a\right)}\\
+F\left(x;a,b\right) & = & \frac{\log\left(x/a\right)}{\log\left(b/a\right)}\\
+G\left(q;a,b\right) & = & a\exp\left(q\log\left(b/a\right)\right)=a\left(\frac{b}{a}\right)^{q}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+d & = & \log\left(a/b\right)\\
+\mu & = & \frac{a-b}{d}\\
+\mu_{2} & = & \mu\frac{a+b}{2}-\mu^{2}=\frac{\left(a-b\right)\left[a\left(d-2\right)+b\left(d+2\right)\right]}{2d^{2}}\\
+\gamma_{1} & = & \frac{\sqrt{2}\left[12d\left(a-b\right)^{2}+d^{2}\left(a^{2}\left(2d-9\right)+2abd+b^{2}\left(2d+9\right)\right)\right]}{3d\sqrt{a-b}\left[a\left(d-2\right)+b\left(d+2\right)\right]^{3/2}}\\
+\gamma_{2} & = & \frac{-36\left(a-b\right)^{3}+36d\left(a-b\right)^{2}\left(a+b\right)-16d^{2}\left(a^{3}-b^{3}\right)+3d^{3}\left(a^{2}+b^{2}\right)\left(a+b\right)}{3\left(a-b\right)\left[a\left(d-2\right)+b\left(d+2\right)\right]^{2}}-3\\
+m_{d} & = & a\\
+m_{n} & = & \sqrt{ab}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=\frac{1}{2}\log\left(ab\right)+\log\left[\log\left(\frac{b}{a}\right)\right].\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Reciprocal Inverse Gaussian
+\end_layout
+
+\begin_layout Standard
+The pdf is found from the inverse gaussian (IG),
+\begin_inset Formula $f_{RIG}\left(x;\mu\right)=\frac{1}{x^{2}}f_{IG}\left(\frac{1}{x};\mu\right)$
+\end_inset
+
+ defined for
+\begin_inset Formula $x\geq0$
+\end_inset
+
+ as
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f_{IG}\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2x\mu^{2}}\right).\\
+F_{IG}\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f_{RIG}\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x}}\exp\left(-\frac{\left(1-\mu x\right)^{2}}{2x\mu^{2}}\right)\\
+F_{RIG}\left(x;\mu\right) & = & 1-F_{IG}\left(\frac{1}{x},\mu\right)\\
+ & = & 1-\Phi\left(\frac{1}{\sqrt{x}}\frac{1-\mu x}{\mu}\right)-\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{1+\mu x}{\mu}\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Semicircular
+\end_layout
+
+\begin_layout Standard
+Defined on
+\begin_inset Formula $x\in\left[-1,1\right]$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{2}{\pi}\sqrt{1-x^{2}}\\
+F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\left[x\sqrt{1-x^{2}}+\arcsin x\right]\\
+G\left(q\right) & = & F^{-1}\left(q\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+m_{d}=m_{n}=\mu & = & 0\\
+\mu_{2} & = & \frac{1}{4}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & -1\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=0.64472988584940017414.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Studentized Range*
+\end_layout
+
+\begin_layout Section
+Student t
+\end_layout
+
+\begin_layout Standard
+Shape parameter
+\begin_inset Formula $\nu>0.$
+\end_inset
+
+
+\begin_inset Formula $I\left(a,b,x\right)$
+\end_inset
+
+ is the incomplete beta integral and
+\begin_inset Formula $I^{-1}\left(a,b,I\left(a,b,x\right)\right)=x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\nu\right) & = & \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi\nu}\Gamma\left(\frac{\nu}{2}\right)\left[1+\frac{x^{2}}{\nu}\right]^{\frac{\nu+1}{2}}}\\
+F\left(x;\nu\right) & = & \left\{ \begin{array}{ccc}
+\frac{1}{2}I\left(\frac{\nu}{2},\frac{1}{2},\frac{\nu}{\nu+x^{2}}\right) & & x\leq0\\
+1-\frac{1}{2}I\left(\frac{\nu}{2},\frac{1}{2},\frac{\nu}{\nu+x^{2}}\right) & & x\geq0\end{array}\right.\\
+G\left(q;\nu\right) & = & \left\{ \begin{array}{ccc}
+-\sqrt{\frac{\nu}{I^{-1}\left(\frac{\nu}{2},\frac{1}{2},2q\right)}-\nu} & & q\leq\frac{1}{2}\\
+\sqrt{\frac{\nu}{I^{-1}\left(\frac{\nu}{2},\frac{1}{2},2-2q\right)}-\nu} & & q\geq\frac{1}{2}\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+m_{n}=m_{d}=\mu & = & 0\\
+\mu_{2} & = & \frac{\nu}{\nu-2}\quad\nu>2\\
+\gamma_{1} & = & 0\quad\nu>3\\
+\gamma_{2} & = & \frac{6}{\nu-4}\quad\nu>4\end{eqnarray*}
+
+\end_inset
+
+ As
+\begin_inset Formula $\nu\rightarrow\infty,$
+\end_inset
+
+ this distribution approaches the standard normal distribution.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=\frac{1}{4}\log\left(\frac{\pi c\Gamma^{2}\left(\frac{c}{2}\right)}{\Gamma^{2}\left(\frac{c+1}{2}\right)}\right)-\frac{\left(c+1\right)}{4}\left[\Psi\left(\frac{c}{2}\right)-cZ\left(c\right)+\pi\tan\left(\frac{\pi c}{2}\right)+\gamma+2\log2\right]\]
+
+\end_inset
+
+where
+\begin_inset Formula \[
+Z\left(c\right)=\,_{3}F_{2}\left(1,1,1+\frac{c}{2};\frac{3}{2},2;1\right)=\sum_{k=0}^{\infty}\frac{k!}{k+1}\frac{\Gamma\left(\frac{c}{2}+1+k\right)}{\Gamma\left(\frac{c}{2}+1\right)}\frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(\frac{3}{2}+k\right)}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Student Z
+\end_layout
+
+\begin_layout Standard
+The student Z distriubtion is defined over all space with one shape parameter
+
+\begin_inset Formula $\nu>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\nu\right) & = & \frac{\Gamma\left(\frac{\nu}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{\nu-1}{2}\right)}\left(1+x^{2}\right)^{-\nu/2}\\
+F\left(x;\nu\right) & = & \left\{ \begin{array}{ccc}
+Q\left(x;\nu\right) & & x\leq0\\
+1-Q\left(x;\nu\right) & & x\geq0\end{array}\right.\\
+Q\left(x;\nu\right) & = & \frac{\left|x\right|^{1-n}\Gamma\left(\frac{n}{2}\right)\,_{2}F_{1}\left(\frac{n-1}{2},\frac{n}{2};\frac{n+1}{2};-\frac{1}{x^{2}}\right)}{2\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}\end{eqnarray*}
+
+\end_inset
+
+Interesting moments are
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 0\\
+\sigma^{2} & = & \frac{1}{\nu-3}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & \frac{6}{\nu-5}.\end{eqnarray*}
+
+\end_inset
+
+ The moment generating function is
+\begin_inset Formula \[
+\theta\left(t\right)=2\sqrt{\left|\frac{t}{2}\right|^{\nu-1}}\frac{K_{\left(n-1\right)/2}\left(\left|t\right|\right)}{\Gamma\left(\frac{\nu-1}{2}\right)}.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Symmetric Power*
+\end_layout
+
+\begin_layout Section
+Triangular
+\end_layout
+
+\begin_layout Standard
+One shape parameter
+\begin_inset Formula $c\in[0,1]$
+\end_inset
+
+ giving the distance to the peak as a percentage of the total extent of
+ the non-zero portion.
+ The location parameter is the start of the non-zero portion, and the scale-para
+meter is the width of the non-zero portion.
+ In standard form we have
+\begin_inset Formula $x\in\left[0,1\right].$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \left\{ \begin{array}{ccc}
+2\frac{x}{c} & & x<c\\
+2\frac{1-x}{1-c} & & x\geq c\end{array}\right.\\
+F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
+\frac{x^{2}}{c} & & x<c\\
+\frac{x^{2}-2x+c}{c-1} & & x\geq c\end{array}\right.\\
+G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
+\sqrt{cq} & & q<c\\
+1-\sqrt{\left(1-c\right)\left(1-q\right)} & & q\geq c\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{c}{3}+\frac{1}{3}\\
+\mu_{2} & = & \frac{1-c+c^{2}}{18}\\
+\gamma_{1} & = & \frac{\sqrt{2}\left(2c-1\right)\left(c+1\right)\left(c-2\right)}{5\left(1-c+c^{2}\right)^{3/2}}\\
+\gamma_{2} & = & -\frac{3}{5}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+h\left(X\right) & = & \log\left(\frac{1}{2}\sqrt{e}\right)\\
+ & \approx & -0.19314718055994530942.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Truncated Exponential
+\end_layout
+
+\begin_layout Standard
+This is an exponential distribution defined only over a certain region
+\begin_inset Formula $0<x<B$
+\end_inset
+
+.
+ In standard form this is
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;B\right) & = & \frac{e^{-x}}{1-e^{-B}}\\
+F\left(x;B\right) & = & \frac{1-e^{-x}}{1-e^{-B}}\\
+G\left(q;B\right) & = & -\log\left(1-q+qe^{-B}\right)\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\Gamma\left(1+n\right)-\Gamma\left(1+n,B\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+h\left[X\right]=\log\left(e^{B}-1\right)+\frac{1+e^{B}\left(B-1\right)}{1-e^{B}}.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Truncated Normal
+\end_layout
+
+\begin_layout Standard
+A normal distribution restricted to lie within a certain range given by
+ two parameters
+\begin_inset Formula $A$
+\end_inset
+
+ and
+\begin_inset Formula $B$
+\end_inset
+
+.
+ Notice that this
+\begin_inset Formula $A$
+\end_inset
+
+ and
+\begin_inset Formula $B$
+\end_inset
+
+ correspond to the bounds on
+\begin_inset Formula $x$
+\end_inset
+
+ in standard form.
+ For
+\begin_inset Formula $x\in\left[A,B\right]$
+\end_inset
+
+ we get
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;A,B\right) & = & \frac{\phi\left(x\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
+F\left(x;A,B\right) & = & \frac{\Phi\left(x\right)-\Phi\left(A\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
+G\left(q;A,B\right) & = & \Phi^{-1}\left[q\Phi\left(B\right)+\Phi\left(A\right)\left(1-q\right)\right]\end{eqnarray*}
+
+\end_inset
+
+ where
+\begin_inset Formula \begin{eqnarray*}
+\phi\left(x\right) & = & \frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\\
+\Phi\left(x\right) & = & \int_{-\infty}^{x}\phi\left(u\right)du.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
+\mu_{2} & = & 1+\frac{A\phi\left(A\right)-B\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}-\left(\frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\right)^{2}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Tukey-Lambda
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;\lambda\right) & = & F^{\prime}\left(x;\lambda\right)=\frac{1}{G^{\prime}\left(F\left(x;\lambda\right);\lambda\right)}=\frac{1}{F^{\lambda-1}\left(x;\lambda\right)+\left[1-F\left(x;\lambda\right)\right]^{\lambda-1}}\\
+F\left(x;\lambda\right) & = & G^{-1}\left(x;\lambda\right)\\
+G\left(p;\lambda\right) & = & \frac{p^{\lambda}-\left(1-p\right)^{\lambda}}{\lambda}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 0\\
+\mu_{2} & = & \int_{0}^{1}G^{2}\left(p;\lambda\right)dp\\
+ & = & \frac{2\Gamma\left(\lambda+\frac{3}{2}\right)-\lambda4^{-\lambda}\sqrt{\pi}\Gamma\left(\lambda\right)\left(1-2\lambda\right)}{\lambda^{2}\left(1+2\lambda\right)\Gamma\left(\lambda+\frac{3}{2}\right)}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & \frac{\mu_{4}}{\mu_{2}^{2}}-3\\
+\mu_{4} & = & \frac{3\Gamma\left(\lambda\right)\Gamma\left(\lambda+\frac{1}{2}\right)2^{-2\lambda}}{\lambda^{3}\Gamma\left(2\lambda+\frac{3}{2}\right)}+\frac{2}{\lambda^{4}\left(1+4\lambda\right)}\\
+ & & -\frac{2\sqrt{3}\Gamma\left(\lambda\right)2^{-6\lambda}3^{3\lambda}\Gamma\left(\lambda+\frac{1}{3}\right)\Gamma\left(\lambda+\frac{2}{3}\right)}{\lambda^{3}\Gamma\left(2\lambda+\frac{3}{2}\right)\Gamma\left(\lambda+\frac{1}{2}\right)}.\end{eqnarray*}
+
+\end_inset
+
+ Notice that the
+\begin_inset Formula $\lim_{\lambda\rightarrow0}G\left(p;\lambda\right)=\log\left(p/\left(1-p\right)\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+h\left[X\right] & = & \int_{0}^{1}\log\left[G^{\prime}\left(p\right)\right]dp\\
+ & = & \int_{0}^{1}\log\left[p^{\lambda-1}+\left(1-p\right)^{\lambda-1}\right]dp.\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Uniform
+\end_layout
+
+\begin_layout Standard
+Standard form
+\begin_inset Formula $x\in\left(0,1\right).$
+\end_inset
+
+ In general form, the lower limit is
+\begin_inset Formula $L,$
+\end_inset
+
+ the upper limit is
+\begin_inset Formula $S+L.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & 1\\
+F\left(x\right) & = & x\\
+G\left(q\right) & = & q\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{1}{2}\\
+\mu_{2} & = & \frac{1}{12}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & -\frac{6}{5}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=0\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Von Mises
+\end_layout
+
+\begin_layout Standard
+Defined for
+\begin_inset Formula $x\in\left[-\pi,\pi\right]$
+\end_inset
+
+ with shape parameter
+\begin_inset Formula $b>0$
+\end_inset
+
+.
+ Note, the PDF and CDF functions are periodic and are always defined over
+
+\begin_inset Formula $x\in\left[-\pi,\pi\right]$
+\end_inset
+
+ regardless of the location parameter.
+ Thus, if an input beyond this range is given, it is converted to the equivalent
+ angle in this range.
+ For values of
+\begin_inset Formula $b<100$
+\end_inset
+
+ the PDF and CDF formulas below are used.
+ Otherwise, a normal approximation with variance
+\begin_inset Formula $1/b$
+\end_inset
+
+ is used.
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;b\right) & = & \frac{e^{b\cos x}}{2\pi I_{0}\left(b\right)}\\
+F\left(x;b\right) & = & \frac{1}{2}+\frac{x}{2\pi}+\sum_{k=1}^{\infty}\frac{I_{k}\left(b\right)\sin\left(kx\right)}{I_{0}\left(b\right)\pi k}\\
+G\left(q;b\right) & = & F^{-1}\left(x;b\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 0\\
+\mu_{2} & = & \int_{-\pi}^{\pi}x^{2}f\left(x;b\right)dx\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & \frac{\int_{-\pi}^{\pi}x^{4}f\left(x;b\right)dx}{\mu_{2}^{2}}-3\end{eqnarray*}
+
+\end_inset
+
+ This can be used for defining circular variance.
+
+\end_layout
+
+\begin_layout Section
+Wald
+\end_layout
+
+\begin_layout Standard
+Special case of the Inverse Normal with shape parameter set to
+\begin_inset Formula $1.0$
+\end_inset
+
+.
+ Defined for
+\begin_inset Formula $x>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+f\left(x\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2x}\right).\\
+F\left(x\right) & = & \Phi\left(\frac{x-1}{\sqrt{x}}\right)+\exp\left(2\right)\Phi\left(-\frac{x+1}{\sqrt{x}}\right)\\
+G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & 1\\
+\mu_{2} & = & 1\\
+\gamma_{1} & = & 3\\
+\gamma_{2} & = & 15\\
+m_{d} & = & \frac{1}{2}\left(\sqrt{13}-3\right)\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Wishart*
+\end_layout
+
+\begin_layout Section
+Wrapped Cauchy
+\end_layout
+
+\begin_layout Standard
+For
+\begin_inset Formula $x\in\left[0,2\pi\right]$
+\end_inset
+
+
+\begin_inset Formula $c\in\left(0,1\right)$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+f\left(x;c\right) & = & \frac{1-c^{2}}{2\pi\left(1+c^{2}-2c\cos x\right)}\\
+g_{c}\left(x\right) & = & \frac{1}{\pi}\arctan\left[\frac{1+c}{1-c}\tan\left(\frac{x}{2}\right)\right]\\
+r_{c}\left(q\right) & = & 2\arctan\left[\frac{1-c}{1+c}\tan\left(\pi q\right)\right]\\
+F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
+g_{c}\left(x\right) & & 0\leq x<\pi\\
+1-g_{c}\left(2\pi-x\right) & & \pi\leq x\leq2\pi\end{array}\right.\\
+G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
+r_{c}\left(q\right) & & 0\leq q<\frac{1}{2}\\
+2\pi-r_{c}\left(1-q\right) & & \frac{1}{2}\leq q\leq1\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).\]
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
Added: trunk/doc/source/tutorial/stats/discrete.lyx
===================================================================
--- trunk/doc/source/tutorial/stats/discrete.lyx (rev 0)
+++ trunk/doc/source/tutorial/stats/discrete.lyx 2010-06-01 04:09:28 UTC (rev 6462)
@@ -0,0 +1,1218 @@
+#LyX 1.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 221
+\textclass article
+\language english
+\inputencoding auto
+\fontscheme default
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize Default
+\paperpackage a4
+\use_geometry 1
+\use_amsmath 1
+\use_natbib 0
+\use_numerical_citations 0
+\paperorientation portrait
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 1in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\quotes_times 2
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+
+\layout Title
+
+Discrete Statistical Distributions
+\layout Standard
+
+Discrete random variables take on only a countable number of values.
+ The commonly used distributions are included in SciPy and described in
+ this document.
+ Each discrete distribution can take one extra integer parameter:
+\begin_inset Formula $L.$
+\end_inset
+
+ The relationship between the general distribution and the standard one
+ is
+\begin_inset Formula \[
+p\left(x\right)=p_{0}\left(x-L\right)\]
+
+\end_inset
+
+ which allows for shifting of the input.
+ When a distribution generator is initialized, the discrete distribution
+ can either specify the beginning and ending (integer) values
+\begin_inset Formula $a$
+\end_inset
+
+ and
+\begin_inset Formula $b$
+\end_inset
+
+ which must be such that
+\begin_inset Formula \[
+p_{0}\left(x\right)=0\quad x<a\textrm{ or }x>b\]
+
+\end_inset
+
+ in which case, it is assumed that the pdf function is specified on the
+ integers
+\begin_inset Formula $a+mk\leq b$
+\end_inset
+
+ where
+\begin_inset Formula $k$
+\end_inset
+
+ is a non-negative integer (
+\begin_inset Formula $0,1,2,\ldots$
+\end_inset
+
+) and
+\begin_inset Formula $m$
+\end_inset
+
+ is a positive integer multiplier.
+ Alternatively, the two lists
+\begin_inset Formula $x_{k}$
+\end_inset
+
+ and
+\begin_inset Formula $p\left(x_{k}\right)$
+\end_inset
+
+ can be provided directly in which case a dictionary is set up internally
+ to evaulate probabilities and generate random variates.
+
+\layout Subsection
+
+Probability Mass Function (PMF)
+\layout Standard
+
+The probability mass function of a random variable X is defined as the probabili
+ty that the random variable takes on a particular value.
+
+\begin_inset Formula \[
+p\left(x_{k}\right)=P\left[X=x_{k}\right]\]
+
+\end_inset
+
+ This is also sometimes called the probability density function, although
+ technically
+\begin_inset Formula \[
+f\left(x\right)=\sum_{k}p\left(x_{k}\right)\delta\left(x-x_{k}\right)\]
+
+\end_inset
+
+ is the probability density function for a discrete distribution
+\begin_inset Foot
+collapsed false
+
+\layout Standard
+
+Note that we will be using
+\begin_inset Formula $p$
+\end_inset
+
+ to represent the probability mass function and a parameter (a probability).
+ The usage should be obvious from context.
+
+\end_inset
+
+.
+
+\layout Subsection
+
+Cumulative Distribution Function (CDF)
+\layout Standard
+
+The cumulative distribution function is
+\begin_inset Formula \[
+F\left(x\right)=P\left[X\leq x\right]=\sum_{x_{k}\leq x}p\left(x_{k}\right)\]
+
+\end_inset
+
+ and is also useful to be able to compute.
+ Note that
+\begin_inset Formula \[
+F\left(x_{k}\right)-F\left(x_{k-1}\right)=p\left(x_{k}\right)\]
+
+\end_inset
+
+
+\layout Subsection
+
+Survival Function
+\layout Standard
+
+The survival function is just
+\begin_inset Formula \[
+S\left(x\right)=1-F\left(x\right)=P\left[X>k\right]\]
+
+\end_inset
+
+ the probability that the random variable is strictly larger than
+\begin_inset Formula $k$
+\end_inset
+
+.
+
+\layout Subsection
+
+Percent Point Function (Inverse CDF)
+\layout Standard
+
+The percent point function is the inverse of the cumulative distribution
+ function and is
+\begin_inset Formula \[
+G\left(q\right)=F^{-1}\left(q\right)\]
+
+\end_inset
+
+ for discrete distributions, this must be modified for cases where there
+ is no
+\begin_inset Formula $x_{k}$
+\end_inset
+
+ such that
+\begin_inset Formula $F\left(x_{k}\right)=q.$
+\end_inset
+
+ In these cases we choose
+\begin_inset Formula $G\left(q\right)$
+\end_inset
+
+ to be the smallest value
+\begin_inset Formula $x_{k}=G\left(q\right)$
+\end_inset
+
+ for which
+\begin_inset Formula $F\left(x_{k}\right)\geq q$
+\end_inset
+
+.
+ If
+\begin_inset Formula $q=0$
+\end_inset
+
+ then we define
+\begin_inset Formula $G\left(0\right)=a-1$
+\end_inset
+
+.
+ This definition allows random variates to be defined in the same way as
+ with continuous rv's using the inverse cdf on a uniform distribution to
+ generate random variates.
+
+\layout Subsection
+
+Inverse survival function
+\layout Standard
+
+The inverse survival function is the inverse of the survival function
+\begin_inset Formula \[
+Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)\]
+
+\end_inset
+
+ and is thus the smallest non-negative integer
+\begin_inset Formula $k$
+\end_inset
+
+ for which
+\begin_inset Formula $F\left(k\right)\geq1-\alpha$
+\end_inset
+
+ or the smallest non-negative integer
+\begin_inset Formula $k$
+\end_inset
+
+ for which
+\begin_inset Formula $S\left(k\right)\leq\alpha.$
+\end_inset
+
+
+\layout Subsection
+
+Hazard functions
+\layout Standard
+
+If desired, the hazard function and the cumulative hazard function could
+ be defined as
+\begin_inset Formula \[
+h\left(x_{k}\right)=\frac{p\left(x_{k}\right)}{1-F\left(x_{k}\right)}\]
+
+\end_inset
+
+ and
+\begin_inset Formula \[
+H\left(x\right)=\sum_{x_{k}\leq x}h\left(x_{k}\right)=\sum_{x_{k}\leq x}\frac{F\left(x_{k}\right)-F\left(x_{k-1}\right)}{1-F\left(x_{k}\right)}.\]
+
+\end_inset
+
+
+\layout Subsection
+
+Moments
+\layout Standard
+
+Non-central moments are defined using the PDF
+\begin_inset Formula \[
+\mu_{m}^{\prime}=E\left[X^{m}\right]=\sum_{k}x_{k}^{m}p\left(x_{k}\right).\]
+
+\end_inset
+
+ Central moments are computed similarly
+\begin_inset Formula $\mu=\mu_{1}^{\prime}$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu_{m}=E\left[\left(X-\mu\right)^{2}\right] & = & \sum_{k}\left(x_{k}-\mu\right)^{m}p\left(x_{k}\right)\\
+ & = & \sum_{k=0}^{m}\left(-1\right)^{m-k}\left(\begin{array}{c}
+m\\
+k\end{array}\right)\mu^{m-k}\mu_{k}^{\prime}\end{eqnarray*}
+
+\end_inset
+
+ The mean is the first moment
+\begin_inset Formula \[
+\mu=\mu_{1}^{\prime}=E\left[X\right]=\sum_{k}x_{k}p\left(x_{k}\right)\]
+
+\end_inset
+
+ the variance is the second central moment
+\begin_inset Formula \[
+\mu_{2}=E\left[\left(X-\mu\right)^{2}\right]=\sum_{x_{k}}x_{k}^{2}p\left(x_{k}\right)-\mu^{2}.\]
+
+\end_inset
+
+Skewness is defined as
+\begin_inset Formula \[
+\gamma_{1}=\frac{\mu_{3}}{\mu_{2}^{3/2}}\]
+
+\end_inset
+
+ while (Fisher) kurtosis is
+\begin_inset Formula \[
+\gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,\]
+
+\end_inset
+
+ so that a normal distribution has a kurtosis of zero.
+
+\layout Subsection
+
+Moment generating function
+\layout Standard
+
+The moment generating funtion is defined as
+\begin_inset Formula \[
+M_{X}\left(t\right)=E\left[e^{Xt}\right]=\sum_{x_{k}}e^{x_{k}t}p\left(x_{k}\right)\]
+
+\end_inset
+
+ Moments are found as the derivatives of the moment generating function
+ evaluated at
+\begin_inset Formula $0.$
+\end_inset
+
+
+\layout Subsection
+
+Fitting data
+\layout Standard
+
+To fit data to a distribution, maximizing the likelihood function is common.
+ Alternatively, some distributions have well-known minimum variance unbiased
+ estimators.
+ These will be chosen by default, but the likelihood function will always
+ be available for minimizing.
+
+\layout Standard
+
+If
+\begin_inset Formula $f_{i}\left(k;\boldsymbol{\theta}\right)$
+\end_inset
+
+ is the PDF of a random-variable where
+\begin_inset Formula $\boldsymbol{\theta}$
+\end_inset
+
+ is a vector of parameters (
+\emph on
+e.g.
+
+\begin_inset Formula $L$
+\end_inset
+
+
+\emph default
+and
+\begin_inset Formula $S$
+\end_inset
+
+), then for a collection of
+\begin_inset Formula $N$
+\end_inset
+
+ independent samples from this distribution, the joint distribution the
+ random vector
+\begin_inset Formula $\mathbf{k}$
+\end_inset
+
+ is
+\begin_inset Formula \[
+f\left(\mathbf{k};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f_{i}\left(k_{i};\boldsymbol{\theta}\right).\]
+
+\end_inset
+
+ The maximum likelihood estimate of the parameters
+\begin_inset Formula $\boldsymbol{\theta}$
+\end_inset
+
+ are the parameters which maximize this function with
+\begin_inset Formula $\mathbf{x}$
+\end_inset
+
+ fixed and given by the data:
+\begin_inset Formula \begin{eqnarray*}
+\hat{\boldsymbol{\theta}} & = & \arg\max_{\boldsymbol{\theta}}f\left(\mathbf{k};\boldsymbol{\theta}\right)\\
+ & = & \arg\min_{\boldsymbol{\theta}}l_{\mathbf{k}}\left(\boldsymbol{\theta}\right).\end{eqnarray*}
+
+\end_inset
+
+ Where
+\begin_inset Formula \begin{eqnarray*}
+l_{\mathbf{k}}\left(\boldsymbol{\theta}\right) & = & -\sum_{i=1}^{N}\log f\left(k_{i};\boldsymbol{\theta}\right)\\
+ & = & -N\overline{\log f\left(k_{i};\boldsymbol{\theta}\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\layout Subsection
+
+Standard notation for mean
+\layout Standard
+
+We will use
+\begin_inset Formula \[
+\overline{y\left(\mathbf{x}\right)}=\frac{1}{N}\sum_{i=1}^{N}y\left(x_{i}\right)\]
+
+\end_inset
+
+ where
+\begin_inset Formula $N$
+\end_inset
+
+ should be clear from context.
+
+\layout Subsection
+
+Combinations
+\layout Standard
+
+Note that
+\begin_inset Formula \[
+k!=k\cdot\left(k-1\right)\cdot\left(k-2\right)\cdot\cdots\cdot1=\Gamma\left(k+1\right)\]
+
+\end_inset
+
+ and has special cases of
+\begin_inset Formula \begin{eqnarray*}
+0! & \equiv & 1\\
+k! & \equiv & 0\quad k<0\end{eqnarray*}
+
+\end_inset
+
+ and
+\begin_inset Formula \[
+\left(\begin{array}{c}
+n\\
+k\end{array}\right)=\frac{n!}{\left(n-k\right)!k!}.\]
+
+\end_inset
+
+ If
+\begin_inset Formula $n<0$
+\end_inset
+
+ or
+\begin_inset Formula $k<0$
+\end_inset
+
+ or
+\begin_inset Formula $k>n$
+\end_inset
+
+ we define
+\begin_inset Formula $\left(\begin{array}{c}
+n\\
+k\end{array}\right)=0$
+\end_inset
+
+
+\layout Section
+
+Bernoulli
+\layout Standard
+
+A Bernoulli random variable of parameter
+\begin_inset Formula $p$
+\end_inset
+
+ takes one of only two values
+\begin_inset Formula $X=0$
+\end_inset
+
+ or
+\begin_inset Formula $X=1$
+\end_inset
+
+.
+ The probability of success (
+\begin_inset Formula $X=1$
+\end_inset
+
+) is
+\begin_inset Formula $p$
+\end_inset
+
+, and the probability of failure (
+\begin_inset Formula $X=0$
+\end_inset
+
+) is
+\begin_inset Formula $1-p.$
+\end_inset
+
+ It can be thought of as a binomial random variable with
+\begin_inset Formula $n=1$
+\end_inset
+
+.
+ The PMF is
+\begin_inset Formula $p\left(k\right)=0$
+\end_inset
+
+ for
+\begin_inset Formula $k\neq0,1$
+\end_inset
+
+ and
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;p\right) & = & \begin{cases}
+1-p & k=0\\
+p & k=1\end{cases}\\
+F\left(x;p\right) & = & \begin{cases}
+0 & x<0\\
+1-p & 0\le x<1\\
+1 & 1\leq x\end{cases}\\
+G\left(q;p\right) & = & \begin{cases}
+0 & 0\leq q<1-p\\
+1 & 1-p\leq q\leq1\end{cases}\\
+\mu & = & p\\
+\mu_{2} & = & p\left(1-p\right)\\
+\gamma_{3} & = & \frac{1-2p}{\sqrt{p\left(1-p\right)}}\\
+\gamma_{4} & = & \frac{1-6p\left(1-p\right)}{p\left(1-p\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \[
+M\left(t\right)=1-p\left(1-e^{t}\right)\]
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \[
+\mu_{m}^{\prime}=p\]
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \[
+h\left[X\right]=p\log p+\left(1-p\right)\log\left(1-p\right)\]
+
+\end_inset
+
+
+\layout Section
+
+Binomial
+\layout Standard
+
+A binomial random variable with parameters
+\begin_inset Formula $\left(n,p\right)$
+\end_inset
+
+ can be described as the sum of
+\begin_inset Formula $n$
+\end_inset
+
+ independent Bernoulli random variables of parameter
+\begin_inset Formula $p;$
+\end_inset
+
+
+\begin_inset Formula \[
+Y=\sum_{i=1}^{n}X_{i}.\]
+
+\end_inset
+
+ Therefore, this random variable counts the number of successes in
+\begin_inset Formula $n$
+\end_inset
+
+ independent trials of a random experiment where the probability of success
+ is
+\begin_inset Formula $p.$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;n,p\right) & = & \left(\begin{array}{c}
+n\\
+k\end{array}\right)p^{k}\left(1-p\right)^{n-k}\,\, k\in\left\{ 0,1,\ldots n\right\} ,\\
+F\left(x;n,p\right) & = & \sum_{k\leq x}\left(\begin{array}{c}
+n\\
+k\end{array}\right)p^{k}\left(1-p\right)^{n-k}=I_{1-p}\left(n-\left\lfloor x\right\rfloor ,\left\lfloor x\right\rfloor +1\right)\quad x\geq0\end{eqnarray*}
+
+\end_inset
+
+ where the incomplete beta integral is
+\begin_inset Formula \[
+I_{x}\left(a,b\right)=\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}\int_{0}^{x}t^{a-1}\left(1-t\right)^{b-1}dt.\]
+
+\end_inset
+
+ Now
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & np\\
+\mu_{2} & = & np\left(1-p\right)\\
+\gamma_{1} & = & \frac{1-2p}{\sqrt{np\left(1-p\right)}}\\
+\gamma_{2} & = & \frac{1-6p\left(1-p\right)}{np\left(1-p\right)}.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\left[1-p\left(1-e^{t}\right)\right]^{n}\]
+
+\end_inset
+
+
+\layout Section
+
+Boltzmann (truncated Planck)
+\layout Standard
+
+
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;N,\lambda\right) & = & \frac{1-e^{-\lambda}}{1-e^{-\lambda N}}\exp\left(-\lambda k\right)\quad k\in\left\{ 0,1,\ldots,N-1\right\} \\
+F\left(x;N,\lambda\right) & = & \left\{ \begin{array}{cc}
+0 & x<0\\
+\frac{1-\exp\left[-\lambda\left(\left\lfloor x\right\rfloor +1\right)\right]}{1-\exp\left(-\lambda N\right)} & 0\leq x\leq N-1\\
+1 & x\geq N-1\end{array}\right.\\
+G\left(q,\lambda\right) & = & \left\lceil -\frac{1}{\lambda}\log\left[1-q\left(1-e^{-\lambda N}\right)\right]-1\right\rceil \end{eqnarray*}
+
+\end_inset
+
+ Define
+\begin_inset Formula $z=e^{-\lambda}$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{z}{1-z}-\frac{Nz^{N}}{1-z^{N}}\\
+\mu_{2} & = & \frac{z}{\left(1-z\right)^{2}}-\frac{N^{2}z^{N}}{\left(1-z^{N}\right)^{2}}\\
+\gamma_{1} & = & \frac{z\left(1+z\right)\left(\frac{1-z^{N}}{1-z}\right)^{3}-N^{3}z^{N}\left(1+z^{N}\right)}{\left[z\left(\frac{1-z^{N}}{1-z}\right)^{2}-N^{2}z^{N}\right]^{3/2}}\\
+\gamma_{2} & = & \frac{z\left(1+4z+z^{2}\right)\left(\frac{1-z^{N}}{1-z}\right)^{4}-N^{4}z^{N}\left(1+4z^{N}+z^{2N}\right)}{\left[z\left(\frac{1-z^{N}}{1-z}\right)^{2}-N^{2}z^{N}\right]^{2}}\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \[
+M\left(t\right)=\frac{1-e^{N\left(t-\lambda\right)}}{1-e^{t-\lambda}}\frac{1-e^{-\lambda}}{1-e^{-\lambda N}}\]
+
+\end_inset
+
+
+\layout Section
+
+Planck (discrete exponential)
+\layout Standard
+
+Named Planck because of its relationship to the black-body problem he solved.
+
+\layout Standard
+
+
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;\lambda\right) & = & \left(1-e^{-\lambda}\right)e^{-\lambda k}\quad k\lambda\geq0\\
+F\left(x;\lambda\right) & = & 1-e^{-\lambda\left(\left\lfloor x\right\rfloor +1\right)}\quad x\lambda\geq0\\
+G\left(q;\lambda\right) & = & \left\lceil -\frac{1}{\lambda}\log\left[1-q\right]-1\right\rceil .\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \frac{1}{e^{\lambda}-1}\\
+\mu_{2} & = & \frac{e^{-\lambda}}{\left(1-e^{-\lambda}\right)^{2}}\\
+\gamma_{1} & = & 2\cosh\left(\frac{\lambda}{2}\right)\\
+\gamma_{2} & = & 4+2\cosh\left(\lambda\right)\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \[
+M\left(t\right)=\frac{1-e^{-\lambda}}{1-e^{t-\lambda}}\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=\frac{\lambda e^{-\lambda}}{1-e^{-\lambda}}-\log\left(1-e^{-\lambda}\right)\]
+
+\end_inset
+
+
+\layout Section
+
+Poisson
+\layout Standard
+
+The Poisson random variable counts the number of successes in
+\begin_inset Formula $n$
+\end_inset
+
+ independent Bernoulli trials in the limit as
+\begin_inset Formula $n\rightarrow\infty$
+\end_inset
+
+ and
+\begin_inset Formula $p\rightarrow0$
+\end_inset
+
+ where the probability of success in each trial is
+\begin_inset Formula $p$
+\end_inset
+
+ and
+\begin_inset Formula $np=\lambda\geq0$
+\end_inset
+
+ is a constant.
+ It can be used to approximate the Binomial random variable or in it's own
+ right to count the number of events that occur in the interval
+\begin_inset Formula $\left[0,t\right]$
+\end_inset
+
+ for a process satisfying certain
+\begin_inset Quotes eld
+\end_inset
+
+sparsity
+\begin_inset Quotes erd
+\end_inset
+
+ constraints.
+ The functions are
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;\lambda\right) & = & e^{-\lambda}\frac{\lambda^{k}}{k!}\quad k\geq0,\\
+F\left(x;\lambda\right) & = & \sum_{n=0}^{\left\lfloor x\right\rfloor }e^{-\lambda}\frac{\lambda^{n}}{n!}=\frac{1}{\Gamma\left(\left\lfloor x\right\rfloor +1\right)}\int_{\lambda}^{\infty}t^{\left\lfloor x\right\rfloor }e^{-t}dt,\\
+\mu & = & \lambda\\
+\mu_{2} & = & \lambda\\
+\gamma_{1} & = & \frac{1}{\sqrt{\lambda}}\\
+\gamma_{2} & = & \frac{1}{\lambda}.\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \[
+M\left(t\right)=\exp\left[\lambda\left(e^{t}-1\right)\right].\]
+
+\end_inset
+
+
+\layout Section
+
+Geometric
+\layout Standard
+
+The geometric random variable with parameter
+\begin_inset Formula $p\in\left(0,1\right)$
+\end_inset
+
+ can be defined as the number of trials required to obtain a success where
+ the probability of success on each trial is
+\begin_inset Formula $p$
+\end_inset
+
+.
+ Thus,
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;p\right) & = & \left(1-p\right)^{k-1}p\quad k\geq1\\
+F\left(x;p\right) & = & 1-\left(1-p\right)^{\left\lfloor x\right\rfloor }\quad x\geq1\\
+G\left(q;p\right) & = & \left\lceil \frac{\log\left(1-q\right)}{\log\left(1-p\right)}\right\rceil \\
+\mu & = & \frac{1}{p}\\
+\mu_{2} & = & \frac{1-p}{p^{2}}\\
+\gamma_{1} & = & \frac{2-p}{\sqrt{1-p}}\\
+\gamma_{2} & = & \frac{p^{2}-6p+6}{1-p}.\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \begin{eqnarray*}
+M\left(t\right) & = & \frac{p}{e^{-t}-\left(1-p\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\layout Section
+
+Negative Binomial
+\layout Standard
+
+The negative binomial random variable with parameters
+\begin_inset Formula $n$
+\end_inset
+
+ and
+\begin_inset Formula $p\in\left(0,1\right)$
+\end_inset
+
+ can be defined as the number of
+\emph on
+extra
+\emph default
+independent trials (beyond
+\begin_inset Formula $n$
+\end_inset
+
+) required to accumulate a total of
+\begin_inset Formula $n$
+\end_inset
+
+ successes where the probability of a success on each trial is
+\begin_inset Formula $p.$
+\end_inset
+
+ Equivalently, this random variable is the number of failures encoutered
+ while accumulating
+\begin_inset Formula $n$
+\end_inset
+
+ successes during independent trials of an experiment that succeeds with
+ probability
+\begin_inset Formula $p.$
+\end_inset
+
+ Thus,
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;n,p\right) & = & \left(\begin{array}{c}
+k+n-1\\
+n-1\end{array}\right)p^{n}\left(1-p\right)^{k}\quad k\geq0\\
+F\left(x;n,p\right) & = & \sum_{i=0}^{\left\lfloor x\right\rfloor }\left(\begin{array}{c}
+i+n-1\\
+i\end{array}\right)p^{n}\left(1-p\right)^{i}\quad x\geq0\\
+ & = & I_{p}\left(n,\left\lfloor x\right\rfloor +1\right)\quad x\geq0\\
+\mu & = & n\frac{1-p}{p}\\
+\mu_{2} & = & n\frac{1-p}{p^{2}}\\
+\gamma_{1} & = & \frac{2-p}{\sqrt{n\left(1-p\right)}}\\
+\gamma_{2} & = & \frac{p^{2}+6\left(1-p\right)}{n\left(1-p\right)}.\end{eqnarray*}
+
+\end_inset
+
+ Recall that
+\begin_inset Formula $I_{p}\left(a,b\right)$
+\end_inset
+
+ is the incomplete beta integral.
+
+\layout Section
+
+Hypergeometric
+\layout Standard
+
+The hypergeometric random variable with parameters
+\begin_inset Formula $\left(M,n,N\right)$
+\end_inset
+
+ counts the number of
+\begin_inset Quotes eld
+\end_inset
+
+good
+\begin_inset Quotes erd
+\end_inset
+
+ objects in a sample of size
+\begin_inset Formula $N$
+\end_inset
+
+ chosen without replacement from a population of
+\begin_inset Formula $M$
+\end_inset
+
+ objects where
+\begin_inset Formula $n$
+\end_inset
+
+ is the number of
+\begin_inset Quotes eld
+\end_inset
+
+good
+\begin_inset Quotes erd
+\end_inset
+
+ objects in the total population.
+
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;N,n,M\right) & = & \frac{\left(\begin{array}{c}
+n\\
+k\end{array}\right)\left(\begin{array}{c}
+M-n\\
+N-k\end{array}\right)}{\left(\begin{array}{c}
+M\\
+N\end{array}\right)}\quad N-\left(M-n\right)\leq k\leq\min\left(n,N\right)\\
+F\left(x;N,n,M\right) & = & \sum_{k=0}^{\left\lfloor x\right\rfloor }\frac{\left(\begin{array}{c}
+m\\
+k\end{array}\right)\left(\begin{array}{c}
+N-m\\
+n-k\end{array}\right)}{\left(\begin{array}{c}
+N\\
+n\end{array}\right)},\\
+\mu & = & \frac{nN}{M}\\
+\mu_{2} & = & \frac{nN\left(M-n\right)\left(M-N\right)}{M^{2}\left(M-1\right)}\\
+\gamma_{1} & = & \frac{\left(M-2n\right)\left(M-2N\right)}{M-2}\sqrt{\frac{M-1}{nN\left(M-m\right)\left(M-n\right)}}\\
+\gamma_{2} & = & \frac{g\left(N,n,M\right)}{nN\left(M-n\right)\left(M-3\right)\left(M-2\right)\left(N-M\right)}\end{eqnarray*}
+
+\end_inset
+
+ where (defining
+\begin_inset Formula $m=M-n$
+\end_inset
+
+)
+\begin_inset Formula \begin{eqnarray*}
+g\left(N,n,M\right) & = & m^{3}-m^{5}+3m^{2}n-6m^{3}n+m^{4}n+3mn^{2}\\
+ & & -12m^{2}n^{2}+8m^{3}n^{2}+n^{3}-6mn^{3}+8m^{2}n^{3}\\
+ & & +mn^{4}-n^{5}-6m^{3}N+6m^{4}N+18m^{2}nN\\
+ & & -6m^{3}nN+18mn^{2}N-24m^{2}n^{2}N-6n^{3}N\\
+ & & -6mn^{3}N+6n^{4}N+6m^{2}N^{2}-6m^{3}N^{2}-24mnN^{2}\\
+ & & +12m^{2}nN^{2}+6n^{2}N^{2}+12mn^{2}N^{2}-6n^{3}N^{2}.\end{eqnarray*}
+
+\end_inset
+
+
+\layout Section
+
+Zipf (Zeta)
+\layout Standard
+
+A random variable has the zeta distribution (also called the zipf distribution)
+ with parameter
+\begin_inset Formula $\alpha>1$
+\end_inset
+
+ if it's probability mass function is given by
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;\alpha\right) & = & \frac{1}{\zeta\left(\alpha\right)k^{\alpha}}\quad k\geq1\end{eqnarray*}
+
+\end_inset
+
+where
+\begin_inset Formula \[
+\zeta\left(\alpha\right)=\sum_{n=1}^{\infty}\frac{1}{n^{\alpha}}\]
+
+\end_inset
+
+ is the Riemann zeta function.
+ Other functions of this distribution are
+\begin_inset Formula \begin{eqnarray*}
+F\left(x;\alpha\right) & = & \frac{1}{\zeta\left(\alpha\right)}\sum_{k=1}^{\left\lfloor x\right\rfloor }\frac{1}{k^{\alpha}}\\
+\mu & = & \frac{\zeta_{1}}{\zeta_{0}}\quad\alpha>2\\
+\mu_{2} & = & \frac{\zeta_{2}\zeta_{0}-\zeta_{1}^{2}}{\zeta_{0}^{2}}\quad\alpha>3\\
+\gamma_{1} & = & \frac{\zeta_{3}\zeta_{0}^{2}-3\zeta_{0}\zeta_{1}\zeta_{2}+2\zeta_{1}^{3}}{\left[\zeta_{2}\zeta_{0}-\zeta_{1}^{2}\right]^{3/2}}\quad\alpha>4\\
+\gamma_{2} & = & \frac{\zeta_{4}\zeta_{0}^{3}-4\zeta_{3}\zeta_{1}\zeta_{0}^{2}+12\zeta_{2}\zeta_{1}^{2}\zeta_{0}-6\zeta_{1}^{4}-3\zeta_{2}^{2}\zeta_{0}^{2}}{\left(\zeta_{2}\zeta_{0}-\zeta_{1}^{2}\right)^{2}}.\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \begin{eqnarray*}
+M\left(t\right) & = & \frac{\textrm{Li}_{\alpha}\left(e^{t}\right)}{\zeta\left(\alpha\right)}\end{eqnarray*}
+
+\end_inset
+
+where
+\begin_inset Formula $\zeta_{i}=\zeta\left(\alpha-i\right)$
+\end_inset
+
+ and
+\begin_inset Formula $\textrm{Li}_{n}\left(z\right)$
+\end_inset
+
+ is the
+\begin_inset Formula $n^{\textrm{th}}$
+\end_inset
+
+ polylogarithm function of
+\begin_inset Formula $z$
+\end_inset
+
+ defined as
+\begin_inset Formula \[
+\textrm{Li}_{n}\left(z\right)\equiv\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\left.M^{\left(n\right)}\left(t\right)\right|_{t=0}=\left.\frac{\textrm{Li}_{\alpha-n}\left(e^{t}\right)}{\zeta\left(a\right)}\right|_{t=0}=\frac{\zeta\left(\alpha-n\right)}{\zeta\left(\alpha\right)}\]
+
+\end_inset
+
+
+\layout Section
+
+Logarithmic (Log-Series, Series)
+\layout Standard
+
+The logarimthic distribution with parameter
+\begin_inset Formula $p$
+\end_inset
+
+ has a probability mass function with terms proportional to the Taylor series
+ expansion of
+\begin_inset Formula $\log\left(1-p\right)$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;p\right) & = & -\frac{p^{k}}{k\log\left(1-p\right)}\quad k\geq1\\
+F\left(x;p\right) & = & -\frac{1}{\log\left(1-p\right)}\sum_{k=1}^{\left\lfloor x\right\rfloor }\frac{p^{k}}{k}=1+\frac{p^{1+\left\lfloor x\right\rfloor }\Phi\left(p,1,1+\left\lfloor x\right\rfloor \right)}{\log\left(1-p\right)}\end{eqnarray*}
+
+\end_inset
+
+where
+\begin_inset Formula \[
+\Phi\left(z,s,a\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{\left(a+k\right)^{s}}\]
+
+\end_inset
+
+ is the Lerch Transcendent.
+ Also define
+\begin_inset Formula $r=\log\left(1-p\right)$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & -\frac{p}{\left(1-p\right)r}\\
+\mu_{2} & = & -\frac{p\left[p+r\right]}{\left(1-p\right)^{2}r^{2}}\\
+\gamma_{1} & = & -\frac{2p^{2}+3pr+\left(1+p\right)r^{2}}{r\left(p+r\right)\sqrt{-p\left(p+r\right)}}r\\
+\gamma_{2} & = & -\frac{6p^{3}+12p^{2}r+p\left(4p+7\right)r^{2}+\left(p^{2}+4p+1\right)r^{3}}{p\left(p+r\right)^{2}}.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+M\left(t\right) & = & -\frac{1}{\log\left(1-p\right)}\sum_{k=1}^{\infty}\frac{e^{tk}p^{k}}{k}\\
+ & = & \frac{\log\left(1-pe^{t}\right)}{\log\left(1-p\right)}\end{eqnarray*}
+
+\end_inset
+
+ Thus,
+\begin_inset Formula \[
+\mu_{n}^{\prime}=\left.M^{\left(n\right)}\left(t\right)\right|_{t=0}=\left.\frac{\textrm{Li}_{1-n}\left(pe^{t}\right)}{\log\left(1-p\right)}\right|_{t=0}=-\frac{\textrm{Li}_{1-n}\left(p\right)}{\log\left(1-p\right)}.\]
+
+\end_inset
+
+
+\layout Section
+
+Discrete Uniform (randint)
+\layout Standard
+
+The discrete uniform distribution with parameters
+\begin_inset Formula $\left(a,b\right)$
+\end_inset
+
+ constructs a random variable that has an equal probability of being any
+ one of the integers in the half-open range
+\begin_inset Formula $[a,b).$
+\end_inset
+
+ If
+\begin_inset Formula $a$
+\end_inset
+
+ is not given it is assumed to be zero and the only parameter is
+\begin_inset Formula $b.$
+\end_inset
+
+ Therefore,
+\begin_inset Formula \begin{eqnarray*}
+p\left(k;a,b\right) & = & \frac{1}{b-a}\quad a\leq k<b\\
+F\left(x;a,b\right) & = & \frac{\left\lfloor x\right\rfloor -a}{b-a}\quad a\leq x\leq b\\
+G\left(q;a,b\right) & = & \left\lceil q\left(b-a\right)+a\right\rceil \\
+\mu & = & \frac{b+a-1}{2}\\
+\mu_{2} & = & \frac{\left(b-a-1\right)\left(b-a+1\right)}{12}\\
+\gamma_{1} & = & 0\\
+\gamma_{2} & = & -\frac{6}{5}\frac{\left(b-a\right)^{2}+1}{\left(b-a-1\right)\left(b-a+1\right)}.\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \begin{eqnarray*}
+M\left(t\right) & = & \frac{1}{b-a}\sum_{k=a}^{b-1}e^{tk}\\
+ & = & \frac{e^{bt}-e^{at}}{\left(b-a\right)\left(e^{t}-1\right)}\end{eqnarray*}
+
+\end_inset
+
+
+\layout Section
+
+Discrete Laplacian
+\layout Standard
+
+Defined over all integers for
+\begin_inset Formula $a>0$
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+p\left(k\right) & = & \tanh\left(\frac{a}{2}\right)e^{-a\left|k\right|},\\
+F\left(x\right) & = & \left\{ \begin{array}{cc}
+\frac{e^{a\left(\left\lfloor x\right\rfloor +1\right)}}{e^{a}+1} & \left\lfloor x\right\rfloor <0,\\
+1-\frac{e^{-a\left\lfloor x\right\rfloor }}{e^{a}+1} & \left\lfloor x\right\rfloor \geq0.\end{array}\right.\\
+G\left(q\right) & = & \left\{ \begin{array}{cc}
+\left\lceil \frac{1}{a}\log\left[q\left(e^{a}+1\right)\right]-1\right\rceil & q<\frac{1}{1+e^{-a}},\\
+\left\lceil -\frac{1}{a}\log\left[\left(1-q\right)\left(1+e^{a}\right)\right]\right\rceil & q\geq\frac{1}{1+e^{-a}}.\end{array}\right.\end{eqnarray*}
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+M\left(t\right) & = & \tanh\left(\frac{a}{2}\right)\sum_{k=-\infty}^{\infty}e^{tk}e^{-a\left|k\right|}\\
+ & = & C\left(1+\sum_{k=1}^{\infty}e^{-\left(t+a\right)k}+\sum_{1}^{\infty}e^{\left(t-a\right)k}\right)\\
+ & = & \tanh\left(\frac{a}{2}\right)\left(1+\frac{e^{-\left(t+a\right)}}{1-e^{-\left(t+a\right)}}+\frac{e^{t-a}}{1-e^{t-a}}\right)\\
+ & = & \frac{\tanh\left(\frac{a}{2}\right)\sinh a}{\cosh a-\cosh t}.\end{eqnarray*}
+
+\end_inset
+
+ Thus,
+\begin_inset Formula \[
+\mu_{n}^{\prime}=M^{\left(n\right)}\left(0\right)=\left[1+\left(-1\right)^{n}\right]\textrm{Li}_{-n}\left(e^{-a}\right)\]
+
+\end_inset
+
+ where
+\begin_inset Formula $\textrm{Li}_{-n}\left(z\right)$
+\end_inset
+
+ is the polylogarithm function of order
+\begin_inset Formula $-n$
+\end_inset
+
+ evaluated at
+\begin_inset Formula $z.$
+\end_inset
+
+
+\begin_inset Formula \[
+h\left[X\right]=-\log\left(\tanh\left(\frac{a}{2}\right)\right)+\frac{a}{\sinh a}\]
+
+\end_inset
+
+
+\layout Section
+
+Discrete Gaussian*
+\layout Standard
+
+Defined for all
+\begin_inset Formula $\mu$
+\end_inset
+
+ and
+\begin_inset Formula $\lambda>0$
+\end_inset
+
+ and
+\begin_inset Formula $k$
+\end_inset
+
+
+\begin_inset Formula \[
+p\left(k;\mu,\lambda\right)=\frac{1}{Z\left(\lambda\right)}\exp\left[-\lambda\left(k-\mu\right)^{2}\right]\]
+
+\end_inset
+
+ where
+\begin_inset Formula \[
+Z\left(\lambda\right)=\sum_{k=-\infty}^{\infty}\exp\left[-\lambda k^{2}\right]\]
+
+\end_inset
+
+
+\begin_inset Formula \begin{eqnarray*}
+\mu & = & \mu\\
+\mu_{2} & = & -\frac{\partial}{\partial\lambda}\log Z\left(\lambda\right)\\
+ & = & G\left(\lambda\right)e^{-\lambda}\end{eqnarray*}
+
+\end_inset
+
+ where
+\begin_inset Formula $G\left(0\right)\rightarrow\infty$
+\end_inset
+
+ and
+\begin_inset Formula $G\left(\infty\right)\rightarrow2$
+\end_inset
+
+ with a minimum less than 2 near
+\begin_inset Formula $\lambda=1$
+\end_inset
+
+
+\begin_inset Formula \[
+G\left(\lambda\right)=\frac{1}{Z\left(\lambda\right)}\sum_{k=-\infty}^{\infty}k^{2}\exp\left[-\lambda\left(k+1\right)\left(k-1\right)\right]\]
+
+\end_inset
+
+
+\the_end
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