[Scipy-svn] r4766 - trunk/scipy/stats

scipy-svn@scip... scipy-svn@scip...
Fri Oct 3 13:57:21 CDT 2008


Author: oliphant
Date: 2008-10-03 13:57:20 -0500 (Fri, 03 Oct 2008)
New Revision: 4766

Modified:
   trunk/scipy/stats/continuous.lyx
   trunk/scipy/stats/distributions.py
Log:
Improve docstring of lognorm a bit.

Modified: trunk/scipy/stats/continuous.lyx
===================================================================
--- trunk/scipy/stats/continuous.lyx	2008-10-03 14:31:41 UTC (rev 4765)
+++ trunk/scipy/stats/continuous.lyx	2008-10-03 18:57:20 UTC (rev 4766)
@@ -1,18 +1,19 @@
-#LyX 1.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 221
+#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 245
+\begin_document
+\begin_header
 \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
+\spacing single
+\papersize default
+\use_geometry true
+\use_amsmath 2
+\cite_engine basic
+\use_bibtopic false
 \paperorientation portrait
 \leftmargin 1in
 \topmargin 1in
@@ -23,43 +24,49 @@
 \paragraph_separation indent
 \defskip medskip
 \quotes_language english
-\quotes_times 2
 \papercolumns 1
 \papersides 1
 \paperpagestyle default
+\tracking_changes false
+\output_changes true
+\end_header
 
-\layout Title
+\begin_body
 
+\begin_layout Title
 Continuous Statistical Distributions
-\layout Section
+\end_layout
 
+\begin_layout Section
 Overview
-\layout Standard
+\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 
+\end_inset
 
  and 
 \begin_inset Formula $S=1.0.$
-\end_inset 
+\end_inset
 
  The nonstandard forms can be obtained for the various functions using (note
  
 \begin_inset Formula $U$
-\end_inset 
+\end_inset
 
  is a standard uniform random variate).
  
-\layout Standard
-\align center 
+\end_layout
 
-\size small 
+\begin_layout Standard
+\align center
 
-\begin_inset  Tabular
+\size small
+\begin_inset Tabular
 <lyxtabular version="3" rows="16" columns="3">
 <features>
 <column alignment="center" valignment="top" leftline="true" width="0pt">
@@ -69,591 +76,612 @@
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Function Name
+\end_layout
 
-Function Name
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Standard Function
+\end_layout
 
-Standard Function
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Transformation
+\end_layout
 
-Transformation
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Cumulative Distribution Function (CDF)
+\end_layout
 
-Cumulative Distribution Function (CDF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $F\left(x\right)$
+\end_inset
 
 
-\begin_inset Formula $F\left(x\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
 
 
-\begin_inset Formula $F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Probability Density Function (PDF)
+\end_layout
 
-Probability Density Function (PDF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $f\left(x\right)=F^{\prime}\left(x\right)$
+\end_inset
 
 
-\begin_inset Formula $f\left(x\right)=F^{\prime}\left(x\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\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
 
 
-\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 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Percent Point Function (PPF)
+\end_layout
 
-Percent Point Function (PPF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $G\left(q\right)=F^{-1}\left(q\right)$
+\end_inset
 
 
-\begin_inset Formula $G\left(q\right)=F^{-1}\left(q\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $G\left(q;L,S\right)=L+SG\left(q\right)$
+\end_inset
 
 
-\begin_inset Formula $G\left(q;L,S\right)=L+SG\left(q\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Probability Sparsity Function (PSF)
+\end_layout
 
-Probability Sparsity Function (PSF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
-
-
+\begin_layout Standard
 \begin_inset Formula $g\left(q\right)=G^{\prime}\left(q\right)$
-\end_inset 
+\end_inset
 
  
-\end_inset 
+\end_layout
+
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $g\left(q;L,S\right)=Sg\left(q\right)$
+\end_inset
 
 
-\begin_inset Formula $g\left(q;L,S\right)=Sg\left(q\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Hazard Function (HF)
+\end_layout
 
-Hazard Function (HF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$
+\end_inset
 
 
-\begin_inset Formula $h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\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
 
 
-\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 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Cumulative Hazard Functon (CHF)
+\end_layout
 
-Cumulative Hazard Functon (CHF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
-
-
+\begin_layout Standard
 \begin_inset Formula $H_{a}\left(x\right)=$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula $\log\frac{1}{1-F\left(x\right)}$
-\end_inset 
+\end_inset
 
 
-\end_inset 
+\end_layout
+
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\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
 
 
-\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 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Survival Function (SF)
+\end_layout
 
-Survival Function (SF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $S\left(x\right)=1-F\left(x\right)$
+\end_inset
 
 
-\begin_inset Formula $S\left(x\right)=1-F\left(x\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
+\end_inset
 
 
-\begin_inset Formula $S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Inverse Survival Function (ISF)
+\end_layout
 
-Inverse Survival Function (ISF)
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$
+\end_inset
 
 
-\begin_inset Formula $Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
+\end_inset
 
 
-\begin_inset Formula $Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Moment Generating Function (MGF) 
+\end_layout
 
-Moment Generating Function (MGF) 
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $M_{Y}\left(t\right)=E\left[e^{Yt}\right]$
+\end_inset
 
 
-\begin_inset Formula $M_{Y}\left(t\right)=E\left[e^{Yt}\right]$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
+\end_inset
 
 
-\begin_inset Formula $M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+Random Variates
+\end_layout
 
-Random Variates
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $Y=G\left(U\right)$
+\end_inset
 
 
-\begin_inset Formula $Y=G\left(U\right)$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $X=L+SY$
+\end_inset
 
 
-\begin_inset Formula $X=L+SY$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+(Differential) Entropy
+\end_layout
 
-(Differential) Entropy
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$
+\end_inset
 
 
-\begin_inset Formula $h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $h\left[X\right]=h\left[Y\right]+\log S$
+\end_inset
 
 
-\begin_inset Formula $h\left[X\right]=h\left[Y\right]+\log S$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+(Non-central) Moments
+\end_layout
 
-(Non-central) Moments
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\mu_{n}^{\prime}=E\left[Y^{n}\right]$
+\end_inset
 
 
-\begin_inset Formula $\mu_{n}^{\prime}=E\left[Y^{n}\right]$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
-
-
+\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_inset
 
 
-\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
 
-\layout Standard
+\begin_layout Standard
+Central Moments 
+\end_layout
 
-Central Moments 
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$
+\end_inset
 
 
-\begin_inset Formula $\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
+\end_inset
 
 
-\begin_inset Formula $E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+mean (mode, median), var
+\end_layout
 
-mean (mode, median), var
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\mu,\,\mu_{2}$
+\end_inset
 
 
-\begin_inset Formula $\mu,\,\mu_{2}$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
-
-
+\begin_layout Standard
 \begin_inset Formula $L+S\mu,\, S^{2}\mu_{2}$
-\end_inset 
+\end_inset
 
   
-\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
 
-\layout Standard
+\begin_layout Standard
+skewness, kurtosis
+\end_layout
 
-skewness, kurtosis
-\end_inset 
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
-
-
+\begin_layout Standard
 \begin_inset Formula $\gamma_{1}=\frac{\mu_{3}}{\left(\mu_{2}\right)^{3/2}},\,$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula $\gamma_{2}=\frac{\mu_{4}}{\left(\mu_{2}\right)^{2}}-3$
-\end_inset 
+\end_inset
 
  
-\end_inset 
+\end_layout
+
+\end_inset
 </cell>
 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 \begin_inset Text
 
-\layout Standard
+\begin_layout Standard
+\begin_inset Formula $\gamma_{1},\,\gamma_{2}$
+\end_inset
 
 
-\begin_inset Formula $\gamma_{1},\,\gamma_{2}$
-\end_inset 
+\end_layout
 
-
-\end_inset 
+\end_inset
 </cell>
 </row>
 </lyxtabular>
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-\SpecialChar ~
+\begin_layout Standard
+\InsetSpace ~
 
-\layout Subsection
+\end_layout
 
+\begin_layout Subsection
 Moments
-\layout Standard
+\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 
+\end_inset
 
  Note, that these can always be computed using the PPF.
  Substitute 
 \begin_inset Formula $x=G\left(q\right)$
-\end_inset 
+\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 
+\end_inset
 
  which may be easier to compute numerically.
  Note that 
 \begin_inset Formula $q=F\left(x\right)$
-\end_inset 
+\end_inset
 
  so that 
 \begin_inset Formula $dq=f\left(x\right)dx.$
-\end_inset 
+\end_inset
 
  Central moments are computed similarly 
 \begin_inset Formula $\mu=\mu_{1}^{\prime}$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -663,7 +691,7 @@
 n\\
 k\end{array}\right)\left(-\mu\right)^{k}\mu_{n-k}^{\prime}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  In particular 
 \begin_inset Formula \begin{eqnarray*}
@@ -672,157 +700,162 @@
 \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 
+\end_inset
 
  Skewness is defined as 
 \begin_inset Formula \[
 \gamma_{1}=\sqrt{\beta_{1}}=\frac{\mu_{3}}{\mu_{2}^{3/2}}\]
 
-\end_inset 
+\end_inset
 
  while (Fisher) kurtosis is 
 \begin_inset Formula \[
 \gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,\]
 
-\end_inset 
+\end_inset
 
  so that a normal distribution has a kurtosis of zero.
  
-\layout Subsection
+\end_layout
 
+\begin_layout Subsection
 Median and mode
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 The median, 
 \begin_inset Formula $m_{n}$
-\end_inset 
+\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 
+\end_inset
 
  so that 
 \begin_inset Formula \[
 m_{n}=G\left(\frac{1}{2}\right).\]
 
-\end_inset 
+\end_inset
 
  In addition, the mode, 
 \begin_inset Formula $m_{d}$
-\end_inset 
+\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_inset
 
 
-\layout Subsection
+\end_layout
 
+\begin_layout Subsection
 Fitting data
-\layout Standard
+\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.
  
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 If 
 \begin_inset Formula $f\left(x;\boldsymbol{\theta}\right)$
-\end_inset 
+\end_inset
 
  is the PDF of a random-variable where 
 \begin_inset Formula $\boldsymbol{\theta}$
-\end_inset 
+\end_inset
 
  is a vector of parameters (
-\emph on 
+\emph on
 e.g.
  
 \begin_inset Formula $L$
-\end_inset 
+\end_inset
 
  
-\emph default 
+\emph default
 and 
 \begin_inset Formula $S$
-\end_inset 
+\end_inset
 
 ), then for a collection of 
 \begin_inset Formula $N$
-\end_inset 
+\end_inset
 
  independent samples from this distribution, the joint distribution the
  random vector 
 \begin_inset Formula $\mathbf{x}$
-\end_inset 
+\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 
+\end_inset
 
  The maximum likelihood estimate of the parameters 
 \begin_inset Formula $\boldsymbol{\theta}$
-\end_inset 
+\end_inset
 
  are the parameters which maximize this function with 
 \begin_inset Formula $\mathbf{x}$
-\end_inset 
+\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 
+\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 
+\end_inset
 
  Note that if 
 \begin_inset Formula $\boldsymbol{\theta}$
-\end_inset 
+\end_inset
 
  includes only shape parameters, the location and scale-parameters can be
  fit by replacing 
 \begin_inset Formula $x_{i}$
-\end_inset 
+\end_inset
 
  with 
 \begin_inset Formula $\left(x_{i}-L\right)/S$
-\end_inset 
+\end_inset
 
  in the log-likelihood function adding 
 \begin_inset Formula $N\log S$
-\end_inset 
+\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 
+\end_inset
 
  If desired, sample estimates for 
 \begin_inset Formula $L$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S$
-\end_inset 
+\end_inset
 
  (not necessarily maximum likelihood estimates) can be obtained from samples
  estimates of the mean and variance using 
@@ -830,185 +863,194 @@
 \hat{S} & = & \sqrt{\frac{\hat{\mu}_{2}}{\mu_{2}}}\\
 \hat{L} & = & \hat{\mu}-\hat{S}\mu\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $\mu$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\mu_{2}$
-\end_inset 
+\end_inset
 
  are assumed known as the mean and variance of the
-\series bold 
+\series bold
  untransformed 
-\series default 
+\series default
 distribution (when 
 \begin_inset Formula $L=0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=1$
-\end_inset 
+\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_inset
 
 
-\layout Subsection
+\end_layout
 
+\begin_layout Subsection
 Standard notation for mean
-\layout Standard
+\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 
+\end_inset
 
  where 
 \begin_inset Formula $N$
-\end_inset 
+\end_inset
 
  should be clear from context as the number of samples 
 \begin_inset Formula $x_{i}$
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Alpha 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 One shape parameters 
 \begin_inset Formula $\alpha>0$
-\end_inset 
+\end_inset
 
  (paramter 
 \begin_inset Formula $\beta$
-\end_inset 
+\end_inset
 
  in DATAPLOT is a scale-parameter).
  Standard form is 
 \begin_inset Formula $x>0:$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Anglit
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined over 
 \begin_inset Formula $x\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
 h\left[X\right] & = & 1-\log2\\
  & \approx & 0.30685281944005469058\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Arcsine 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined over 
 \begin_inset Formula $x\in\left(0,1\right)$
-\end_inset 
+\end_inset
 
 .
  To get the JKB definition put 
 \begin_inset Formula $x=\frac{u+1}{2}.$
-\end_inset 
+\end_inset
 
  i.e.
  
 \begin_inset Formula $L=-1$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=2.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1016,20 +1058,20 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \[
 M\left(t\right)=E^{t/2}I_{0}\left(\frac{t}{2}\right)\]
 
-\end_inset 
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1038,45 +1080,47 @@
 \gamma_{1} & = & 0\\
 \gamma_{2} & = & -\frac{3}{2}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]\approx-0.24156447527049044468\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Beta 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Two shape parameters
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 a,b>0\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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)\\
@@ -1088,66 +1132,68 @@
 \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_inset
 
  
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula $f\left(x;a,1\right)$
-\end_inset 
+\end_inset
 
  is also called the Power-function distribution.
-\layout Standard
+\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 
+\end_inset
 
  All of the 
 \begin_inset Formula $x_{i}\in\left[0,1\right]$
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Beta Prime
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined over 
 \begin_inset Formula $0<x<\infty.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $\alpha,\beta>0.$
-\end_inset 
+\end_inset
 
  (Note the CDF evaluation uses Eq.
  3.194.1 on pg.
  313 of Gradshteyn & Ryzhik (sixth edition).
  
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  Therefore, 
 \begin_inset Formula \begin{eqnarray*}
@@ -1157,20 +1203,21 @@
 \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_inset
 
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Bradford
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \begin{eqnarray*}
 c & > & 0\\
 k & = & \log\left(1+c\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1185,37 +1232,38 @@
 m_{d} & = & 0\\
 m_{n} & = & \sqrt{1+c}-1\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $\textrm{Ei}\left(\textrm{z}\right)$
-\end_inset 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Burr
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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}\\
@@ -1230,15 +1278,16 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Cauchy
-\layout Standard
+\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\\
@@ -1246,7 +1295,7 @@
 m_{d} & = & 0\\
 m_{n} & = & 0\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 No finite moments.
  This is the t distribution with one degree of freedom.
@@ -1255,35 +1304,37 @@
 h\left[X\right] & = & \log\left(4\pi\right)\\
  & \approx & 2.5310242469692907930.\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Chi
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Generated by taking the (positive) square-root of chi-squared variates.
-\layout Standard
+\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 
+\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_inset
 
 
-\layout Standard
+\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}\\
@@ -1292,50 +1343,53 @@
 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_inset
 
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Chi-squared
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is the gamma distribution with 
 \begin_inset Formula $L=0.0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=2.0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\alpha=\nu/2$
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $\nu$
-\end_inset 
+\end_inset
 
  is called the degrees of freedom.
  If 
 \begin_inset Formula $Z_{1}\ldots Z_{\nu}$
-\end_inset 
+\end_inset
 
  are all standard normal distributions, then 
 \begin_inset Formula $W=\sum_{k}Z_{k}^{2}$
-\end_inset 
+\end_inset
 
  has (standard) chi-square distribution with 
 \begin_inset Formula $\nu$
-\end_inset 
+\end_inset
 
  degrees of freedom.
  
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 The standard form (most often used in standard form only) is 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -1343,13 +1397,13 @@
 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 
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1359,19 +1413,21 @@
 \gamma_{2} & = & \frac{12}{\nu}\\
 m_{d} & = & \frac{\nu}{2}-1\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Cosine
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Approximation to the normal distribution.
  
-\layout Standard
+\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)\\
@@ -1382,33 +1438,35 @@
 \gamma_{1} & = & 0\\
 \gamma_{2} & = & \frac{-6\left(\pi^{4}-90\right)}{5\left(\pi^{2}-6\right)^{2}}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Double Gamma
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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}
@@ -1418,21 +1476,21 @@
 -\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_inset
 
  
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \begin{eqnarray*}
 \mu=m_{n} & = & 0\\
 \mu_{2} & = & \alpha\left(\alpha+1\right)\\
@@ -1440,24 +1498,28 @@
 \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Doubly Non-central F*
-\layout Section
+\end_layout
 
+\begin_layout Section
 Doubly Non-central t*
-\layout Section
+\end_layout
 
+\begin_layout Section
 Double Weibull
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is a signed form of the Weibull distribution.
-\layout Standard
+\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}
@@ -1467,7 +1529,7 @@
 -\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 
+\end_inset
 
 
 \begin_inset Formula \[
@@ -1475,7 +1537,7 @@
 \Gamma\left(1+\frac{n}{c}\right) & n\textrm{ even}\\
 0 & n\textrm{ odd}\end{cases}\]
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1485,34 +1547,38 @@
 \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Erlang
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is just the Gamma distribution with shape parameter 
 \begin_inset Formula $\alpha=n$
-\end_inset 
+\end_inset
 
  an integer.
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Exponential
-\layout Standard
+\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 
+\end_inset
 
  and the same location and scale parameters.
  The standard form is therefore (
 \begin_inset Formula $x\geq0$
-\end_inset 
+\end_inset
 
 ) 
 \begin_inset Formula \begin{eqnarray*}
@@ -1520,25 +1586,25 @@
 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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 \mu_{n}^{\prime}=n!\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 M\left(t\right)=\frac{1}{1-t}\]
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -1548,31 +1614,33 @@
 \gamma_{2} & = & 6\\
 m_{d} & = & 0\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \[
 h\left[X\right]=1.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Exponentiated Weibull
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Two positive shape parameters 
 \begin_inset Formula $a$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $x\in\left(0,\infty\right)$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1580,22 +1648,24 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Exponential Power
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 One positive shape parameter 
 \begin_inset Formula $b$
-\end_inset 
+\end_inset
 
 .
  Defined for 
 \begin_inset Formula $x\geq0.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1603,84 +1673,88 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Fatigue Life (Birnbaum-Sanders)
-\layout Standard
+\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 
+\end_inset
 
  and reciprocal inverse-Gaussian pdf 
 \begin_inset Formula $\left(\mu=1\right)$
-\end_inset 
+\end_inset
 
 .
  We follow the notation of JKB here with 
 \begin_inset Formula $\beta=S.$
-\end_inset 
+\end_inset
 
  for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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 
+\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_inset
 
  
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Fisk (Log Logistic)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Special case of the Burr distribution with 
 \begin_inset Formula $d=1$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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}\\
@@ -1695,45 +1769,47 @@
 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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]=2-\log c.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Folded Cauchy
-\layout Standard
+\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 
+\end_inset
 
  and the pdf 
 \begin_inset Formula $d\left(x\right)$
-\end_inset 
+\end_inset
 
 ).
  if 
 \begin_inset Formula $Y$
-\end_inset 
+\end_inset
 
  is cauchy then 
 \begin_inset Formula $\left|Y\right|$
-\end_inset 
+\end_inset
 
  is folded cauchy.
  Note that 
 \begin_inset Formula $x\geq0.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1741,54 +1817,57 @@
 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_inset
 
 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 No moments
-\layout Section
+\end_layout
 
+\begin_layout Section
 Folded Normal
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 If 
 \begin_inset Formula $Z$
-\end_inset 
+\end_inset
 
  is Normal with mean 
 \begin_inset Formula $L$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\sigma=S$
-\end_inset 
+\end_inset
 
 , then 
 \begin_inset Formula $\left|Z\right|$
-\end_inset 
+\end_inset
 
  is a folded normal with shape parameter 
 \begin_inset Formula $c=\left|L\right|/S$
-\end_inset 
+\end_inset
 
 , location parameter 
 \begin_inset Formula $0$
-\end_inset 
+\end_inset
 
  and scale parameter 
 \begin_inset Formula $S$
-\end_inset 
+\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 
+\end_inset
 
  Note that 
 \begin_inset Formula $c\geq0$
-\end_inset 
+\end_inset
 
 .
  The standard form of the folded normal is 
@@ -1797,13 +1876,13 @@
 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 
+\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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -1814,38 +1893,40 @@
 \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Fratio (or F)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 .
  The distribution of 
 \begin_inset Formula $\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)$
-\end_inset 
+\end_inset
 
  if 
 \begin_inset Formula $X_{1}$
-\end_inset 
+\end_inset
 
  is chi-squared with 
 \begin_inset Formula $v_{1}$
-\end_inset 
+\end_inset
 
  degrees of freedom and 
 \begin_inset Formula $X_{2}$
-\end_inset 
+\end_inset
 
  is chi-squared with 
 \begin_inset Formula $v_{2}$
-\end_inset 
+\end_inset
 
  degrees of freedom.
  
@@ -1854,7 +1935,7 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -1863,22 +1944,24 @@
 \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_inset
 
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A type of extreme-value distribution with a lower bound.
  Defined for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -1886,13 +1969,13 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \[
 \mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right)\]
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -1903,37 +1986,39 @@
 m_{d} & = & \left(\frac{c}{1+c}\right)^{1/c}\\
 m_{n} & = & G\left(\frac{1}{2};c\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \[
 h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $\gamma$
-\end_inset 
+\end_inset
 
  is Euler's constant and equal to 
 \begin_inset Formula \[
 \gamma\approx0.57721566490153286061.\]
 
-\end_inset 
+\end_inset
 
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined for 
 \begin_inset Formula $x<0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
 .
  
@@ -1942,53 +2027,56 @@
 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_inset
 
 
-\layout Standard
+\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
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $\gamma$
-\end_inset 
+\end_inset
 
  is Euler's constant and equal to 
 \begin_inset Formula \[
 \gamma\approx0.57721566490153286061.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Gamma
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 The standard form for the gamma distribution is 
 \begin_inset Formula $\left(\alpha>0\right)$
-\end_inset 
+\end_inset
 
  valid for 
 \begin_inset Formula $x\geq0$
-\end_inset 
+\end_inset
 
 .
 \begin_inset Formula \begin{eqnarray*}
@@ -1996,13 +2084,13 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \[
 M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}\]
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2012,62 +2100,64 @@
 \gamma_{2} & = & \frac{6}{\alpha}\\
 m_{d} & = & \alpha-1\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  where 
 \begin_inset Formula \[
 \Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Generalized Logistic
-\layout Standard
+\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 
+\end_inset
 
  And 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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)\\
@@ -2076,7 +2166,7 @@
 m_{d} & = & \log c\\
 m_{n} & = & -\log\left(2^{1/c}-1\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  Note that the polygamma function is 
 \begin_inset Formula \begin{eqnarray*}
@@ -2084,42 +2174,44 @@
  & = & \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 
+\end_inset
 
  where 
 \begin_inset Formula $\zeta\left(k,x\right)$
-\end_inset 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Generalized Pareto
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Shape parameter 
 \begin_inset Formula $c\neq0$
-\end_inset 
+\end_inset
 
  and defined for 
 \begin_inset Formula $x\geq0$
-\end_inset 
+\end_inset
 
  for all 
 \begin_inset Formula $c$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $x<\frac{1}{\left|c\right|}$
-\end_inset 
+\end_inset
 
  if 
 \begin_inset Formula $c$
-\end_inset 
+\end_inset
 
  is negative.
  
@@ -2128,29 +2220,29 @@
 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_inset
 
 
-\layout Standard
+\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_inset
 
  
-\layout Standard
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2159,7 +2251,7 @@
 \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 
+\end_inset
 
  Thus,
 \begin_inset Formula \begin{eqnarray*}
@@ -2168,38 +2260,40 @@
 \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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]=1+c\quad c>0\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Generalized Exponential
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Three positive shape parameters for 
 \begin_inset Formula $x\geq0.$
-\end_inset 
+\end_inset
 
  Note that 
 \begin_inset Formula $a,b,$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c$
-\end_inset 
+\end_inset
 
  are all 
 \begin_inset Formula $>0.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2207,29 +2301,32 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Generalized Extreme Value
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Extreme value distributions with shape parameter 
 \begin_inset Formula $c$
-\end_inset 
+\end_inset
 
 .
  
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 For 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  defined on 
 \begin_inset Formula $-\infty<x\leq1/c.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2237,7 +2334,7 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \[
@@ -2245,7 +2342,7 @@
 n\\
 k\end{array}\right)\left(-1\right)^{k}\Gamma\left(ck+1\right)\quad cn>-1\]
 
-\end_inset 
+\end_inset
 
  So,
 \begin_inset Formula \begin{eqnarray*}
@@ -2254,19 +2351,19 @@
 \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 
+\end_inset
 
  For 
 \begin_inset Formula $c<0$
-\end_inset 
+\end_inset
 
  defined on 
 \begin_inset Formula $\frac{1}{c}\leq x<\infty.$
-\end_inset 
+\end_inset
 
  For 
 \begin_inset Formula $c=0$
-\end_inset 
+\end_inset
 
  defined over all space 
 \begin_inset Formula \begin{eqnarray*}
@@ -2274,7 +2371,7 @@
 F\left(x;0\right) & = & \exp\left[-e^{-x}\right]\\
 G\left(q;0\right) & = & -\log\left(-\log q\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  This is just the (left-skewed) Gumbel distribution for c=0.
  
@@ -2284,30 +2381,32 @@
 \gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
 \gamma_{2} & = & \frac{12}{5}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Generalized Gamma
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A general probability form that reduces to many common distributions: 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $a>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c\neq0.$
-\end_inset 
+\end_inset
 
  
-\layout Standard
+\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}
@@ -2316,13 +2415,13 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \[
 \mu_{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)}\]
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2332,47 +2431,49 @@
 \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 
+\end_inset
 
  Special cases are Weibull 
 \begin_inset Formula $\left(a=1\right)$
-\end_inset 
+\end_inset
 
 , half-normal 
 \begin_inset Formula $\left(a=1/2,c=2\right)$
-\end_inset 
+\end_inset
 
  and ordinary gamma distributions 
 \begin_inset Formula $c=1.$
-\end_inset 
+\end_inset
 
  If 
 \begin_inset Formula $c=-1$
-\end_inset 
+\end_inset
 
  then it is the inverted gamma distribution.
  
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Generalized Half-Logistic
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 For 
 \begin_inset Formula $x\in\left[0,1/c\right]$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  we have 
 \begin_inset Formula \begin{eqnarray*}
@@ -2380,34 +2481,36 @@
 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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Gilbrat
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Special case of the log-normal with 
 \begin_inset Formula $\sigma=1$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=1.0$
-\end_inset 
+\end_inset
 
  (typically also 
 \begin_inset Formula $L=0.0$
-\end_inset 
+\end_inset
 
 ) 
 \begin_inset Formula \begin{eqnarray*}
@@ -2415,7 +2518,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2424,58 +2527,60 @@
 \gamma_{1} & = & \sqrt{e-1}\left(2+e\right)\\
 \gamma_{2} & = & e^{4}+2e^{3}+3e^{2}-6\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Gompertz (Truncated Gumbel)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 For 
 \begin_inset Formula $x\geq0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
 .
  In JKB the two shape parameters 
 \begin_inset Formula $b,a$
-\end_inset 
+\end_inset
 
  are reduced to the single shape-parameter 
 \begin_inset Formula $c=b/a$
-\end_inset 
+\end_inset
 
 .
  As 
 \begin_inset Formula $a$
-\end_inset 
+\end_inset
 
  is just a scale parameter when 
 \begin_inset Formula $a\neq0$
-\end_inset 
+\end_inset
 
 .
  If 
 \begin_inset Formula $a=0,$
-\end_inset 
+\end_inset
 
  the distribution reduces to the exponential distribution scaled by 
 \begin_inset Formula $1/b.$
-\end_inset 
+\end_inset
 
  Thus, the standard form is given as 
 \begin_inset Formula \begin{eqnarray*}
@@ -2483,45 +2588,47 @@
 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_inset
 
 
-\layout Standard
+\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 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 One of a clase of extreme value distributions (right-skewed).
-\layout Standard
+\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 
+\end_inset
 
 
 \begin_inset Formula \[
 M\left(t\right)=\Gamma\left(1-t\right)\]
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2532,80 +2639,83 @@
 m_{d} & = & 0\\
 m_{n} & = & -\log\left(\log2\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]\approx1.0608407169541684911\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Gumbel Left-skewed (for minimum order statistic)
-\layout Standard
+\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 
+\end_inset
 
 
 \begin_inset Formula \[
 M\left(t\right)=\Gamma\left(1+t\right)\]
 
-\end_inset 
+\end_inset
 
  Note, that 
 \begin_inset Formula $\mu$
-\end_inset 
+\end_inset
 
  is negative the mean for the right-skewed distribution.
  Similar for median and mode.
  All other moments are the same.
  
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]\approx1.0608407169541684911.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 HalfCauchy
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 If 
 \begin_inset Formula $Z$
-\end_inset 
+\end_inset
 
  is Hyperbolic Secant distributed then 
 \begin_inset Formula $e^{Z}$
-\end_inset 
+\end_inset
 
  is Half-Cauchy distributed.
  Also, if 
 \begin_inset Formula $W$
-\end_inset 
+\end_inset
 
  is (standard) Cauchy distributed, then 
 \begin_inset Formula $\left|W\right|$
-\end_inset 
+\end_inset
 
  is Half-Cauchy distributed.
  Special case of the Folded Cauchy distribution with 
 \begin_inset Formula $c=0.$
-\end_inset 
+\end_inset
 
  The standard form is 
 \begin_inset Formula \begin{eqnarray*}
@@ -2613,68 +2723,70 @@
 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 
+\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_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  No moments, as the integrals diverge.
  
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 HalfNormal
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is a special case of the chi distribution with 
 \begin_inset Formula $L=a$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=b$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\nu=1.$
-\end_inset 
+\end_inset
 
  This is also a special case of the folded normal with shape parameter 
 \begin_inset Formula $c=0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=S.$
-\end_inset 
+\end_inset
 
  If 
 \begin_inset Formula $Z$
-\end_inset 
+\end_inset
 
  is (standard) normally distributed then, 
 \begin_inset Formula $\left|Z\right|$
-\end_inset 
+\end_inset
 
  is half-normal.
  The standard form is 
@@ -2683,18 +2795,18 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \[
 M\left(t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right)\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \begin{eqnarray*}
 \mu & = & \sqrt{\frac{2}{\pi}}\\
 \mu_{2} & = & 1-\frac{2}{\pi}\\
@@ -2703,40 +2815,42 @@
 m_{d} & = & 0\\
 m_{n} & = & \Phi^{-1}\left(\frac{3}{4}\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Half-Logistic 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 In the limit as 
 \begin_inset Formula $c\rightarrow\infty$
-\end_inset 
+\end_inset
 
  for the generalized half-logistic we have the half-logistic defined over
  
 \begin_inset Formula $x\geq0.$
-\end_inset 
+\end_inset
 
  Also, the distribution of 
 \begin_inset Formula $\left|X\right|$
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $X$
-\end_inset 
+\end_inset
 
  has logistic distribtution.
  
@@ -2745,22 +2859,22 @@
 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_inset
 
  
-\layout Standard
+\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 
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2769,28 +2883,30 @@
 \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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Hyperbolic Secant
-\layout Standard
+\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 
+\end_inset
 
 )
 \begin_inset Formula \begin{eqnarray*}
@@ -2798,13 +2914,13 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \[
 M\left(t\right)=\sec\left(\frac{\pi}{2}t\right)\]
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -2813,30 +2929,30 @@
 0 & n\textrm{ odd}\\
 C_{n/2}\frac{\pi^{n}}{2^{n}} & n\textrm{ even}\end{array}\right.\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $C_{m}$
-\end_inset 
+\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 
+\end_inset
 
 where 
 \begin_inset Formula $B_{2m+1}\left(\frac{1}{4}\right)$
-\end_inset 
+\end_inset
 
  is the Bernoulli polynomial of order 
 \begin_inset Formula $2m+1$
-\end_inset 
+\end_inset
 
  evaluated at 
 \begin_inset Formula $1/4.$
-\end_inset 
+\end_inset
 
  Thus 
 \begin_inset Formula \[
@@ -2844,90 +2960,93 @@
 0 & n\textrm{ odd}\\
 4\left(-1\right)^{n/2-1}\frac{\left(2\pi\right)^{n}}{n+1}B_{n+1}\left(\frac{1}{4}\right) & n\textrm{ even}\end{array}\right.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]=\log\left(2\pi\right).\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Gauss Hypergeometric 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula $x\in\left[0,1\right]$
-\end_inset 
+\end_inset
 
 , 
 \begin_inset Formula $\alpha>0,\,\beta>0$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \[
 C^{-1}=B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)\]
 
-\end_inset 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Inverted Gamma
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Special case of the generalized Gamma distribution with 
 \begin_inset Formula $c=-1$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $a>0$
-\end_inset 
+\end_inset
 
 , 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  
-\layout Standard
+\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 
+\end_inset
 
  
 \begin_inset Formula \[
 \mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n\]
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2936,54 +3055,56 @@
 \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 
+\end_inset
 
 
 \begin_inset Formula \[
 m_{d}=\frac{1}{a+1}\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Inverse Normal (Inverse Gaussian)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 The standard form involves the shape parameter 
 \begin_inset Formula $\mu$
-\end_inset 
+\end_inset
 
  (in most definitions, 
 \begin_inset Formula $L=0.0$
-\end_inset 
+\end_inset
 
  is used).
  (In terms of the regress documentation 
 \begin_inset Formula $\mu=A/B$
-\end_inset 
+\end_inset
 
 ) and 
 \begin_inset Formula $B=S$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $L$
-\end_inset 
+\end_inset
 
  is not a parameter in that distribution.
  A standard form is 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -2991,12 +3112,12 @@
 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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \begin{eqnarray*}
 \mu & = & \mu\\
 \mu_{2} & = & \mu^{3}\\
@@ -3004,80 +3125,83 @@
 \gamma_{2} & = & 15\mu\\
 m_{d} & = & \frac{\mu}{2}\left(\sqrt{9\mu^{2}+4}-3\mu\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is related to the canonical form or JKB 
 \begin_inset Quotes eld
-\end_inset 
+\end_inset
 
 two-parameter
 \begin_inset Quotes erd
-\end_inset 
+\end_inset
 
  inverse Gaussian when written in it's full form with scale parameter 
 \begin_inset Formula $S$
-\end_inset 
+\end_inset
 
  and location parameter 
 \begin_inset Formula $L$
-\end_inset 
+\end_inset
 
  by taking 
 \begin_inset Formula $L=0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S\equiv\lambda,$
-\end_inset 
+\end_inset
 
  then 
 \begin_inset Formula $\mu S$
-\end_inset 
+\end_inset
 
  is equal to 
 \begin_inset Formula $\mu_{2}$
-\end_inset 
+\end_inset
 
  where
-\bar under 
+\bar under
  
-\bar default 
+\bar default
 
 \begin_inset Formula $\mu_{2}$
-\end_inset 
+\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 
+\end_inset
 
  and the kurtosis (
 \begin_inset Formula $\beta_{2}-3$
-\end_inset 
+\end_inset
 
 ) are both functions only of 
 \begin_inset Formula $\mu_{2}/\lambda=\mu S/S=\mu$
-\end_inset 
+\end_inset
 
  as shown here, while the variance and mean of the standard form here are
  transformed appropriately.
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Inverted Weibull
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Shape parameter 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 .
  Then 
@@ -3086,35 +3210,37 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=1+\gamma+\frac{\gamma}{c}-\log\left(c\right)\]
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $\gamma$
-\end_inset 
+\end_inset
 
  is Euler's constant.
-\layout Section
+\end_layout
 
+\begin_layout Section
 Johnson SB
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined for 
 \begin_inset Formula $x\in\left(0,1\right)$
-\end_inset 
+\end_inset
 
  with two shape parameters 
 \begin_inset Formula $a$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $b>0.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3122,25 +3248,27 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Johnson SU
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined for all 
 \begin_inset Formula $x$
-\end_inset 
+\end_inset
 
  with two shape parameters 
 \begin_inset Formula $a$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $b>0$
-\end_inset 
+\end_inset
 
 .
  
@@ -3149,21 +3277,24 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 KSone
-\layout Section
+\end_layout
 
+\begin_layout Section
 KStwo
-\layout Section
+\end_layout
 
+\begin_layout Section
 Laplace (Double Exponential, Bilateral Expoooonential)
-\layout Standard
+\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}
@@ -3173,7 +3304,7 @@
 \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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3182,95 +3313,98 @@
 \gamma_{1} & = & 0\\
 \gamma_{2} & = & 3\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  where 
 \begin_inset Formula $X_{i}$
-\end_inset 
+\end_inset
 
  is a sequence of 
 \begin_inset Formula $N$
-\end_inset 
+\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 
+\end_inset
 
  and the 
 \begin_inset Formula $(N/2+1)\textrm{th}$
-\end_inset 
+\end_inset
 
  order statistic (
-\emph on 
+\emph on
 e.g.
 
-\emph default 
+\emph default
  take the average of these two) when 
 \begin_inset Formula $N$
-\end_inset 
+\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 
+\end_inset
 
  Replace 
 \begin_inset Formula $\hat{L}$
-\end_inset 
+\end_inset
 
  with 
 \begin_inset Formula $L$
-\end_inset 
+\end_inset
 
  if it is known.
  If 
 \begin_inset Formula $L$
-\end_inset 
+\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_inset
 
 .
  
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Left-skewed Lévy
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Special case of Lévy-stable distribution with 
 \begin_inset Formula $\alpha=\frac{1}{2}$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\beta=-1$
-\end_inset 
+\end_inset
 
  the support is 
 \begin_inset Formula $x<0$
-\end_inset 
+\end_inset
 
 .
  In standard form
@@ -3279,26 +3413,28 @@
 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 
+\end_inset
 
 No moments.
-\layout Section
+\end_layout
 
+\begin_layout Section
 Lévy
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A special case of Lévy-stable distributions with 
 \begin_inset Formula $\alpha=\frac{1}{2}$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\beta=1$
-\end_inset 
+\end_inset
 
 .
  In standard form it is defined for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  as 
 \begin_inset Formula \begin{eqnarray*}
@@ -3306,37 +3442,39 @@
 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 
+\end_inset
 
  It has no finite moments.
-\layout Section
+\end_layout
 
+\begin_layout Section
 Logistic (Sech-squared)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A special case of the Generalized Logistic distribution with 
 \begin_inset Formula $c=1.$
-\end_inset 
+\end_inset
 
  Defined for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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}\\
@@ -3345,30 +3483,32 @@
 m_{d} & = & \log1=0\\
 m_{n} & = & -\log\left(2-1\right)=0\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]=1.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Log Double Exponential (Log-Laplace)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined over 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  with 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3382,30 +3522,32 @@
 \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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left[X\right]=\log\left(\frac{2e}{c}\right)\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Log Gamma
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A single shape parameter 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  (Defined for all 
 \begin_inset Formula $x$
-\end_inset 
+\end_inset
 
 ) 
 \begin_inset Formula \begin{eqnarray*}
@@ -3413,13 +3555,13 @@
 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 
+\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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -3428,43 +3570,45 @@
 \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Log Normal (Cobb-Douglass)
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Has one shape parameter 
 \begin_inset Formula $\sigma$
-\end_inset 
+\end_inset
 
 >0.
  (Notice that the 
 \begin_inset Quotes eld
-\end_inset 
+\end_inset
 
 Regress
 \begin_inset Quotes erd
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $A=\log S$
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $S$
-\end_inset 
+\end_inset
 
  is the scale parameter and 
 \begin_inset Formula $A$
-\end_inset 
+\end_inset
 
  is the mean of the underlying normal distribution).
  The standard form is 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -3472,7 +3616,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3481,45 +3625,80 @@
 \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_inset
 
   
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Notice that using JKB notation we have 
 \begin_inset Formula $\theta=L,$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $\zeta=\log S$
-\end_inset 
+\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.
  
-\layout Standard
+\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_inset
 
 
-\layout Section
+\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
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Generalization of the chi distribution.
  Shape parameter is 
 \begin_inset Formula $\nu>0.$
-\end_inset 
+\end_inset
 
  Defined for 
 \begin_inset Formula $x>0.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3527,7 +3706,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3536,70 +3715,75 @@
 \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Noncentral beta*
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined over 
 \begin_inset Formula $x\in\left[0,1\right]$
-\end_inset 
+\end_inset
 
  with 
 \begin_inset Formula $a>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $b>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c\geq0$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Noncentral chi*
-\layout Section
+\end_layout
 
+\begin_layout Section
 Noncentral chi-squared
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 The distribution of 
 \begin_inset Formula $\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}$
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $Z_{i}$
-\end_inset 
+\end_inset
 
  are independent standard normal variables and 
 \begin_inset Formula $\delta_{i}$
-\end_inset 
+\end_inset
 
  are constants.
  
 \begin_inset Formula $\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.$
-\end_inset 
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3607,7 +3791,7 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -3616,68 +3800,72 @@
 \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Noncentral F
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Let 
 \begin_inset Formula $\lambda>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\nu_{1}>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\nu_{2}>0.$
-\end_inset 
+\end_inset
 
  
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Noncentral t
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 The distribution of the ratio 
 \begin_inset Formula \[
 \frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}}\]
 
-\end_inset 
+\end_inset
 
  where 
 \begin_inset Formula $U$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\chi_{\nu}$
-\end_inset 
+\end_inset
 
  are independent and distributed as a standard normal and chi with 
 \begin_inset Formula $\nu$
-\end_inset 
+\end_inset
 
  degrees of freedom.
  Note 
 \begin_inset Formula $\lambda>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\nu>0$
-\end_inset 
+\end_inset
 
 .
  
@@ -3689,77 +3877,81 @@
  &  & \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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Normal
-\layout Standard
+\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_inset
 
 
-\layout Standard
-\align center 
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Maxwell
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is a special case of the Chi distribution with 
 \begin_inset Formula $L=0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=S=\frac{1}{\sqrt{a}}$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\nu=3.$
-\end_inset 
+\end_inset
 
  
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \begin{eqnarray*}
 \mu & = & 2\sqrt{\frac{2}{\pi}}\\
 \mu_{2} & = & 3-\frac{8}{\pi}\\
@@ -3768,37 +3960,39 @@
 m_{d} & = & \sqrt{2}\\
 m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\frac{1}{2}\right)}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=\log\left(\sqrt{\frac{2\pi}{e}}\right)+\gamma.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Mielke's Beta-Kappa
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A generalized F distribution.
  Two shape parameters 
 \begin_inset Formula $\kappa$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\theta$
-\end_inset 
+\end_inset
 
 , and 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 .
  The 
 \begin_inset Formula $\beta$
-\end_inset 
+\end_inset
 
  in the DATAPLOT reference is a scale parameter.
 \begin_inset Formula \begin{eqnarray*}
@@ -3806,107 +4000,112 @@
 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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Pareto
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 For 
 \begin_inset Formula $x\geq1$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $b>0$
-\end_inset 
+\end_inset
 
 .
  Standard form is 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 h\left(X\right)=\frac{1}{c}+1-\log\left(c\right)\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Pareto Second Kind (Lomax)
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula $c>0.$
-\end_inset 
+\end_inset
 
  This is Pareto of the first kind with 
 \begin_inset Formula $L=-1.0$
-\end_inset 
+\end_inset
 
  so 
 \begin_inset Formula $x\geq0$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=\frac{1}{c}+1-\log\left(c\right).\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Power Log Normal
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A generalization of the log-normal distribution 
 \begin_inset Formula $\sigma>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3914,13 +4113,13 @@
 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 
+\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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -3929,21 +4128,23 @@
 \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_inset
 
  This distribution reduces to the log-normal distribution when 
 \begin_inset Formula $c=1.$
-\end_inset 
+\end_inset
 
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Power Normal
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A generalization of the normal distribution, 
 \begin_inset Formula $c>0$
-\end_inset 
+\end_inset
 
  for 
 \begin_inset Formula \begin{eqnarray*}
@@ -3951,13 +4152,13 @@
 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 
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -3966,40 +4167,42 @@
 \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_inset
 
 For 
 \begin_inset Formula $c=1$
-\end_inset 
+\end_inset
 
  this reduces to the normal distribution.
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Power-function 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A special case of the beta distribution with 
 \begin_inset Formula $b=1$
-\end_inset 
+\end_inset
 
 : defined for 
 \begin_inset Formula $x\in\left[0,1\right]$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 a>0\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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}\\
@@ -4010,63 +4213,67 @@
 \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 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=1-\frac{1}{a}-\log\left(a\right)\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 R-distribution
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 A general-purpose distribution with a variety of shapes controlled by 
 \begin_inset Formula $c>0.$
-\end_inset 
+\end_inset
 
  Range of standard distribution is 
 \begin_inset Formula $x\in\left[-1,1\right]$
-\end_inset 
+\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 
+\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 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Rayleigh 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is Chi distribution with 
 \begin_inset Formula $L=0.0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $\nu=2$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $S=S$
-\end_inset 
+\end_inset
 
  (no location parameter is generally used), the mode of the distribution
  is 
 \begin_inset Formula $S.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4074,7 +4281,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4085,69 +4292,73 @@
 m_{d} & = & 1\\
 m_{n} & = & \sqrt{2\log\left(2\right)}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \[
 h\left[X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right).\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \[
 \mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right)\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Rice*
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $b>0$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Reciprocal
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Shape parameters 
 \begin_inset Formula $a,b>0$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $x\in\left[a,b\right]$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4155,7 +4366,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4167,58 +4378,62 @@
 m_{d} & = & a\\
 m_{n} & = & \sqrt{ab}\end{eqnarray*}
 
-\end_inset 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Reciprocal Inverse Gaussian 
-\layout Standard
+\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 
+\end_inset
 
  defined for 
 \begin_inset Formula $x\geq0$
-\end_inset 
+\end_inset
 
  as 
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Semicircular
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined on 
 \begin_inset Formula $x\in\left[-1,1\right]$
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -4226,7 +4441,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4235,39 +4450,42 @@
 \gamma_{1} & = & 0\\
 \gamma_{2} & = & -1\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=0.64472988584940017414.\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Studentized Range*
-\layout Section
+\end_layout
 
+\begin_layout Section
 Student t
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Shape parameter 
 \begin_inset Formula $\nu>0.$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $I\left(a,b,x\right)$
-\end_inset 
+\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_inset
 
  
-\layout Standard
+\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}
@@ -4277,50 +4495,52 @@
 -\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_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  As 
 \begin_inset Formula $\nu\rightarrow\infty,$
-\end_inset 
+\end_inset
 
  this distribution approaches the standard normal distribution.
  
-\layout Standard
+\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 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Student Z
-\layout Standard
+\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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -4330,7 +4550,7 @@
 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 
+\end_inset
 
 Interesting moments are 
 \begin_inset Formula \begin{eqnarray*}
@@ -4339,26 +4559,29 @@
 \gamma_{1} & = & 0\\
 \gamma_{2} & = & \frac{6}{\nu-5}.\end{eqnarray*}
 
-\end_inset 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Symmetric Power*
-\layout Section
+\end_layout
 
+\begin_layout Section
 Triangular
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 One shape parameter 
 \begin_inset Formula $c\in[0,1]$
-\end_inset 
+\end_inset
 
  giving the distance to the peak as a percentage of the total extent of
  the non-zero portion.
@@ -4366,7 +4589,7 @@
 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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4380,7 +4603,7 @@
 \sqrt{cq} &  & q<c\\
 1-\sqrt{\left(1-c\right)\left(1-q\right)} &  & q\geq c\end{array}\right.\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -4389,24 +4612,26 @@
 \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 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Truncated Exponential
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 This is an exponential distribution defined only over a certain region 
 \begin_inset Formula $0<x<B$
-\end_inset 
+\end_inset
 
 .
  In standard form this is 
@@ -4415,55 +4640,57 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \[
 \mu_{n}^{\prime}=\Gamma\left(1+n\right)-\Gamma\left(1+n,B\right)\]
 
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Truncated Normal
-\layout Standard
+\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 
+\end_inset
 
  and 
 \begin_inset Formula $B$
-\end_inset 
+\end_inset
 
 .
  Notice that this 
 \begin_inset Formula $A$
-\end_inset 
+\end_inset
 
  and 
 \begin_inset Formula $B$
-\end_inset 
+\end_inset
 
  correspond to the bounds on 
 \begin_inset Formula $x$
-\end_inset 
+\end_inset
 
  in standard form.
  For 
 \begin_inset Formula $x\in\left[A,B\right]$
-\end_inset 
+\end_inset
 
  we get 
 \begin_inset Formula \begin{eqnarray*}
@@ -4471,35 +4698,36 @@
 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 
+\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 
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Tukey-Lambda
-\layout Standard
+\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 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4511,50 +4739,52 @@
 \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 
+\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_inset
 
 
-\layout Standard
+\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_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Uniform
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Standard form 
 \begin_inset Formula $x\in\left(0,1\right).$
-\end_inset 
+\end_inset
 
  In general form, the lower limit is 
 \begin_inset Formula $L,$
-\end_inset 
+\end_inset
 
  the upper limit is 
 \begin_inset Formula $S+L.$
-\end_inset 
+\end_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
 
 \begin_inset Formula \begin{eqnarray*}
@@ -4563,45 +4793,47 @@
 \gamma_{1} & = & 0\\
 \gamma_{2} & = & -\frac{6}{5}\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=0\]
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Von Mises
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Defined for 
 \begin_inset Formula $x\in\left[-\pi,\pi\right]$
-\end_inset 
+\end_inset
 
  with shape parameter 
 \begin_inset Formula $b>0$
-\end_inset 
+\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 
+\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 
+\end_inset
 
  the PDF and CDF formulas below are used.
  Otherwise, a normal approximation with variance 
 \begin_inset Formula $1/b$
-\end_inset 
+\end_inset
 
  is used.
  
@@ -4610,51 +4842,53 @@
 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_inset
 
 
-\layout Standard
+\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 
+\end_inset
 
  This can be used for defining circular variance.
  
-\layout Section
+\end_layout
 
+\begin_layout Section
 Wald 
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 Special case of the Inverse Normal with shape parameter set to 
 \begin_inset Formula $1.0$
-\end_inset 
+\end_inset
 
 .
  Defined for 
 \begin_inset Formula $x>0$
-\end_inset 
+\end_inset
 
 .
-\layout Standard
+\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_inset
 
 
-\layout Standard
+\end_layout
 
-
+\begin_layout Standard
 \begin_inset Formula \begin{eqnarray*}
 \mu & = & 1\\
 \mu_{2} & = & 1\\
@@ -4662,24 +4896,27 @@
 \gamma_{2} & = & 15\\
 m_{d} & = & \frac{1}{2}\left(\sqrt{13}-3\right)\end{eqnarray*}
 
-\end_inset 
+\end_inset
 
 
-\layout Section
+\end_layout
 
+\begin_layout Section
 Wishart*
-\layout Section
+\end_layout
 
+\begin_layout Section
 Wrapped Cauchy
-\layout Standard
+\end_layout
 
+\begin_layout Standard
 For 
 \begin_inset Formula $x\in\left[0,2\pi\right]$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula $c\in\left(0,1\right)$
-\end_inset 
+\end_inset
 
  
 \begin_inset Formula \begin{eqnarray*}
@@ -4693,19 +4930,22 @@
 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 
+\end_inset
 
 
 \begin_inset Formula \[
 \]
 
-\end_inset 
+\end_inset
 
 
 \begin_inset Formula \[
 h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).\]
 
-\end_inset 
+\end_inset
 
  
-\the_end
+\end_layout
+
+\end_body
+\end_document

Modified: trunk/scipy/stats/distributions.py
===================================================================
--- trunk/scipy/stats/distributions.py	2008-10-03 14:31:41 UTC (rev 4765)
+++ trunk/scipy/stats/distributions.py	2008-10-03 18:57:20 UTC (rev 4766)
@@ -2280,7 +2280,6 @@
 
 ## Log-Laplace  (Log Double Exponential)
 ##
-
 class loglaplace_gen(rv_continuous):
     def _pdf(self, x, c):
         cd2 = c/2.0
@@ -2336,6 +2335,10 @@
 
 lognorm.pdf(x,s) = 1/(s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)
 for x > 0, s > 0.
+
+If log x is normally distributed with mean mu and variance sigma**2,
+then x is log-normally distributed with shape paramter sigma and scale
+parameter exp(mu).
 """
                       )
 



More information about the Scipy-svn mailing list