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\textclass article
\language english
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\author "" 
\end_header

\begin_body

\begin_layout Title
Continuous Statistical Distributions
\end_layout

\begin_layout Section
Overview
\end_layout

\begin_layout Standard
All distributions will have location (L) and Scale (S) parameters along
 with any shape parameters needed, the names for the shape parameters will
 vary.
 Standard form for the distributions will be given where 
\begin_inset Formula $L=0.0$
\end_inset

 and 
\begin_inset Formula $S=1.0.$
\end_inset

 The nonstandard forms can be obtained for the various functions using (note
 
\begin_inset Formula $U$
\end_inset

 is a standard uniform random variate).
 
\end_layout

\begin_layout Standard
\align center

\size small
\begin_inset Tabular
<lyxtabular version="3" rows="16" columns="3">
<features>
<column alignment="center" valignment="top" leftline="true" width="0pt">
<column alignment="center" valignment="top" leftline="true" width="0pt">
<column alignment="center" valignment="top" leftline="true" rightline="true" width="0pt">
<row topline="true" bottomline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Function Name
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Standard Function
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Transformation
\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Cumulative Distribution Function (CDF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $F\left(x\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Probability Density Function (PDF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $f\left(x\right)=F^{\prime}\left(x\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $f\left(x;L,S\right)=\frac{1}{S}f\left(\frac{\left(x-L\right)}{S}\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Percent Point Function (PPF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $G\left(q\right)=F^{-1}\left(q\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $G\left(q;L,S\right)=L+SG\left(q\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Probability Sparsity Function (PSF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $g\left(q\right)=G^{\prime}\left(q\right)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $g\left(q;L,S\right)=Sg\left(q\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Hazard Function (HF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $h_{a}\left(x;L,S\right)=\frac{1}{S}h_{a}\left(\frac{\left(x-L\right)}{S}\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Cumulative Hazard Functon (CHF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $H_{a}\left(x\right)=$
\end_inset


\begin_inset Formula $\log\frac{1}{1-F\left(x\right)}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $H_{a}\left(x;L,S\right)=H_{a}\left(\frac{\left(x-L\right)}{S}\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Survival Function (SF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $S\left(x\right)=1-F\left(x\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Inverse Survival Function (ISF)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Moment Generating Function (MGF) 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $M_{Y}\left(t\right)=E\left[e^{Yt}\right]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Random Variates
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $Y=G\left(U\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $X=L+SY$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
(Differential) Entropy
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $h\left[X\right]=h\left[Y\right]+\log S$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
(Non-central) Moments
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $\mu_{n}^{\prime}=E\left[Y^{n}\right]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $E\left[X^{n}\right]=L^{n}\sum_{k=0}^{N}\left(\begin{array}{c}
n\\
k\end{array}\right)\left(\frac{S}{L}\right)^{k}\mu_{k}^{\prime}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
Central Moments 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
mean (mode, median), var
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $\mu,\,\mu_{2}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $L+S\mu,\, S^{2}\mu_{2}$
\end_inset

  
\end_layout

\end_inset
</cell>
</row>
<row topline="true" bottomline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
skewness, kurtosis
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $\gamma_{1}=\frac{\mu_{3}}{\left(\mu_{2}\right)^{3/2}},\,$
\end_inset


\begin_inset Formula $\gamma_{2}=\frac{\mu_{4}}{\left(\mu_{2}\right)^{2}}-3$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Standard
\begin_inset Formula $\gamma_{1},\,\gamma_{2}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\InsetSpace ~

\end_layout

\begin_layout Subsection
Moments
\end_layout

\begin_layout Standard
Non-central moments are defined using the PDF 
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{-\infty}^{\infty}x^{n}f\left(x\right)dx.\]

\end_inset

 Note, that these can always be computed using the PPF.
 Substitute 
\begin_inset Formula $x=G\left(q\right)$
\end_inset

 in the above equation and get 
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{0}^{1}G^{n}\left(q\right)dq\]

\end_inset

 which may be easier to compute numerically.
 Note that 
\begin_inset Formula $q=F\left(x\right)$
\end_inset

 so that 
\begin_inset Formula $dq=f\left(x\right)dx.$
\end_inset

 Central moments are computed similarly 
\begin_inset Formula $\mu=\mu_{1}^{\prime}$
\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu_{n} & = & \int_{-\infty}^{\infty}\left(x-\mu\right)^{n}f\left(x\right)dx\\
 & = & \int_{0}^{1}\left(G\left(q\right)-\mu\right)^{n}dq\\
 & = & \sum_{k=0}^{n}\left(\begin{array}{c}
n\\
k\end{array}\right)\left(-\mu\right)^{k}\mu_{n-k}^{\prime}\end{eqnarray*}

\end_inset

 In particular 
\begin_inset Formula \begin{eqnarray*}
\mu_{3} & = & \mu_{3}^{\prime}-3\mu\mu_{2}^{\prime}+2\mu^{3}\\
 & = & \mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}\\
\mu_{4} & = & \mu_{4}^{\prime}-4\mu\mu_{3}^{\prime}+6\mu^{2}\mu_{2}^{\prime}-3\mu^{4}\\
 & = & \mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\end{eqnarray*}

\end_inset

 Skewness is defined as 
\begin_inset Formula \[
\gamma_{1}=\sqrt{\beta_{1}}=\frac{\mu_{3}}{\mu_{2}^{3/2}}\]

\end_inset

 while (Fisher) kurtosis is 
\begin_inset Formula \[
\gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,\]

\end_inset

 so that a normal distribution has a kurtosis of zero.
 
\end_layout

\begin_layout Subsection
Median and mode
\end_layout

\begin_layout Standard
The median, 
\begin_inset Formula $m_{n}$
\end_inset

 is defined as the point at which half of the density is on one side and
 half on the other.
 In other words, 
\begin_inset Formula $F\left(m_{n}\right)=\frac{1}{2}$
\end_inset

 so that 
\begin_inset Formula \[
m_{n}=G\left(\frac{1}{2}\right).\]

\end_inset

 In addition, the mode, 
\begin_inset Formula $m_{d}$
\end_inset

, is defined as the value for which the probability density function reaches
 it's peak 
\begin_inset Formula \[
m_{d}=\arg\max_{x}f\left(x\right).\]

\end_inset


\end_layout

\begin_layout Subsection
Fitting data
\end_layout

\begin_layout Standard
To fit data to a distribution, maximizing the likelihood function is common.
 Alternatively, some distributions have well-known minimum variance unbiased
 estimators.
 These will be chosen by default, but the likelihood function will always
 be available for minimizing.
 
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $f\left(x;\boldsymbol{\theta}\right)$
\end_inset

 is the PDF of a random-variable where 
\begin_inset Formula $\boldsymbol{\theta}$
\end_inset

 is a vector of parameters (
\emph on
e.g.
 
\begin_inset Formula $L$
\end_inset


\emph default
 and 
\begin_inset Formula $S$
\end_inset

), then for a collection of 
\begin_inset Formula $N$
\end_inset

 independent samples from this distribution, the joint distribution the
 random vector 
\begin_inset Formula $\mathbf{x}$
\end_inset

 is 
\begin_inset Formula \[
f\left(\mathbf{x};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f\left(x_{i};\boldsymbol{\theta}\right).\]

\end_inset

 The maximum likelihood estimate of the parameters 
\begin_inset Formula $\boldsymbol{\theta}$
\end_inset

 are the parameters which maximize this function with 
\begin_inset Formula $\mathbf{x}$
\end_inset

 fixed and given by the data: 
\begin_inset Formula \begin{eqnarray*}
\boldsymbol{\theta}_{es} & = & \arg\max_{\boldsymbol{\theta}}f\left(\mathbf{x};\boldsymbol{\theta}\right)\\
 & = & \arg\min_{\boldsymbol{\theta}}l_{\mathbf{x}}\left(\boldsymbol{\theta}\right).\end{eqnarray*}

\end_inset

 Where 
\begin_inset Formula \begin{eqnarray*}
l_{\mathbf{x}}\left(\boldsymbol{\theta}\right) & = & -\sum_{i=1}^{N}\log f\left(x_{i};\boldsymbol{\theta}\right)\\
 & = & -N\overline{\log f\left(x_{i};\boldsymbol{\theta}\right)}\end{eqnarray*}

\end_inset

 Note that if 
\begin_inset Formula $\boldsymbol{\theta}$
\end_inset

 includes only shape parameters, the location and scale-parameters can be
 fit by replacing 
\begin_inset Formula $x_{i}$
\end_inset

 with 
\begin_inset Formula $\left(x_{i}-L\right)/S$
\end_inset

 in the log-likelihood function adding 
\begin_inset Formula $N\log S$
\end_inset

 and minimizing, thus 
\begin_inset Formula \begin{eqnarray*}
l_{\mathbf{x}}\left(L,S;\boldsymbol{\theta}\right) & = & N\log S-\sum_{i=1}^{N}\log f\left(\frac{x_{i}-L}{S};\boldsymbol{\theta}\right)\\
 & = & N\log S+l_{\frac{\mathbf{x}-S}{L}}\left(\boldsymbol{\theta}\right)\end{eqnarray*}

\end_inset

 If desired, sample estimates for 
\begin_inset Formula $L$
\end_inset

 and 
\begin_inset Formula $S$
\end_inset

 (not necessarily maximum likelihood estimates) can be obtained from samples
 estimates of the mean and variance using 
\begin_inset Formula \begin{eqnarray*}
\hat{S} & = & \sqrt{\frac{\hat{\mu}_{2}}{\mu_{2}}}\\
\hat{L} & = & \hat{\mu}-\hat{S}\mu\end{eqnarray*}

\end_inset

 where 
\begin_inset Formula $\mu$
\end_inset

 and 
\begin_inset Formula $\mu_{2}$
\end_inset

 are assumed known as the mean and variance of the 
\series bold
untransformed
\series default
 distribution (when 
\begin_inset Formula $L=0$
\end_inset

 and 
\begin_inset Formula $S=1$
\end_inset

) and 
\begin_inset Formula \begin{eqnarray*}
\hat{\mu} & = & \frac{1}{N}\sum_{i=1}^{N}x_{i}=\bar{\mathbf{x}}\\
\hat{\mu}_{2} & = & \frac{1}{N-1}\sum_{i=1}^{N}\left(x_{i}-\hat{\mu}\right)^{2}=\frac{N}{N-1}\overline{\left(\mathbf{x}-\bar{\mathbf{x}}\right)^{2}}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Subsection
Standard notation for mean
\end_layout

\begin_layout Standard
We will use 
\begin_inset Formula \[
\overline{y\left(\mathbf{x}\right)}=\frac{1}{N}\sum_{i=1}^{N}y\left(x_{i}\right)\]

\end_inset

 where 
\begin_inset Formula $N$
\end_inset

 should be clear from context as the number of samples 
\begin_inset Formula $x_{i}$
\end_inset


\end_layout

\begin_layout Section
Alpha 
\end_layout

\begin_layout Standard
One shape parameters 
\begin_inset Formula $\alpha>0$
\end_inset

 (paramter 
\begin_inset Formula $\beta$
\end_inset

 in DATAPLOT is a scale-parameter).
 Standard form is 
\begin_inset Formula $x>0:$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha\right) & = & \frac{1}{x^{2}\Phi\left(\alpha\right)\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{x}\right)^{2}\right)\\
F\left(x;\alpha\right) & = & \frac{\Phi\left(\alpha-\frac{1}{x}\right)}{\Phi\left(\alpha\right)}\\
G\left(q;\alpha\right) & = & \left[\alpha-\Phi^{-1}\left(q\Phi\left(\alpha\right)\right)\right]^{-1}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{1}{\Phi\left(a\right)\sqrt{2\pi}}\int_{0}^{\infty}\frac{e^{xt}}{x^{2}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{x}\right)^{2}\right)dx\]

\end_inset


\end_layout

\begin_layout Standard
No moments?
\begin_inset Formula \[
l_{\mathbf{x}}\left(\alpha\right)=N\log\left[\Phi\left(\alpha\right)\sqrt{2\pi}\right]+2N\overline{\log\mathbf{x}}+\frac{N}{2}\alpha^{2}-\alpha\overline{\mathbf{x}^{-1}}+\frac{1}{2}\overline{\mathbf{x}^{-2}}\]

\end_inset


\end_layout

\begin_layout Section
Anglit
\end_layout

\begin_layout Standard
Defined over 
\begin_inset Formula $x\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \sin\left(2x+\frac{\pi}{2}\right)=\cos\left(2x\right)\\
F\left(x\right) & = & \sin^{2}\left(x+\frac{\pi}{4}\right)\\
G\left(q\right) & = & \arcsin\left(\sqrt{q}\right)-\frac{\pi}{4}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{16}-\frac{1}{2}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & -2\frac{\pi^{4}-96}{\left(\pi^{2}-8\right)^{2}}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & 1-\log2\\
 & \approx & 0.30685281944005469058\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
M\left(t\right) & = & \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\cos\left(2x\right)e^{xt}dx\\
 & = & \frac{4\cosh\left(\frac{\pi t}{4}\right)}{t^{2}+4}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
l_{\mathbf{x}}\left(\cdot\right)=-N\overline{\log\left[\cos\left(2\mathbf{x}\right)\right]}\]

\end_inset


\end_layout

\begin_layout Section
Arcsine 
\end_layout

\begin_layout Standard
Defined over 
\begin_inset Formula $x\in\left(0,1\right)$
\end_inset

.
 To get the JKB definition put 
\begin_inset Formula $x=\frac{u+1}{2}.$
\end_inset

 i.e.
 
\begin_inset Formula $L=-1$
\end_inset

 and 
\begin_inset Formula $S=2.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\pi\sqrt{x\left(1-x\right)}}\\
F\left(x\right) & = & \frac{2}{\pi}\arcsin\left(\sqrt{x}\right)\\
G\left(q\right) & = & \sin^{2}\left(\frac{\pi}{2}q\right)\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
M\left(t\right)=E^{t/2}I_{0}\left(\frac{t}{2}\right)\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu_{n}^{\prime} & = & \frac{1}{\pi}\int_{0}^{1}dx\, x^{n-1/2}\left(1-x\right)^{-1/2}\\
 & = & \frac{1}{\pi}B\left(\frac{1}{2},n+\frac{1}{2}\right)=\frac{\left(2n-1\right)!!}{2^{n}n!}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{1}{2}\\
\mu_{2} & = & \frac{1}{8}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & -\frac{3}{2}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]\approx-0.24156447527049044468\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
l_{\mathbf{x}}\left(\cdot\right)=N\log\pi+\frac{N}{2}\overline{\log\mathbf{x}}+\frac{N}{2}\overline{\log\left(1-\mathbf{x}\right)}\]

\end_inset


\end_layout

\begin_layout Section
Beta 
\end_layout

\begin_layout Standard
Two shape parameters
\end_layout

\begin_layout Standard
\begin_inset Formula \[
a,b>0\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,b\right) & = & \frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}x^{a-1}\left(1-x\right)^{b-1}I_{\left(0,1\right)}\left(x\right)\\
F\left(x;a,b\right) & = & \int_{0}^{x}f\left(y;a,b\right)dy=I\left(x,a,b\right)\\
G\left(\alpha;a,b\right) & = & I^{-1}\left(\alpha;a,b\right)\\
M\left(t\right) & = & \frac{\Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}\,_{1}F_{1}\left(a;a+b;t\right)\\
\mu & = & \frac{a}{a+b}\\
\mu_{2} & = & \frac{ab\left(a+b+1\right)}{\left(a+b\right)^{2}}\\
\gamma_{1} & = & 2\frac{b-a}{a+b+2}\sqrt{\frac{a+b+1}{ab}}\\
\gamma_{2} & = & \frac{6\left(a^{3}+a^{2}\left(1-2b\right)+b^{2}\left(b+1\right)-2ab\left(b+2\right)\right)}{ab\left(a+b+2\right)\left(a+b+3\right)}\\
m_{d} & = & \frac{\left(a-1\right)}{\left(a+b-2\right)}\, a+b\neq2\end{eqnarray*}

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula $f\left(x;a,1\right)$
\end_inset

 is also called the Power-function distribution.
\end_layout

\begin_layout Standard
\begin_inset Formula \[
l_{\mathbf{x}}\left(a,b\right)=-N\log\Gamma\left(a+b\right)+N\log\Gamma\left(a\right)+N\log\Gamma\left(b\right)-N\left(a-1\right)\overline{\log\mathbf{x}}-N\left(b-1\right)\overline{\log\left(1-\mathbf{x}\right)}\]

\end_inset

 All of the 
\begin_inset Formula $x_{i}\in\left[0,1\right]$
\end_inset


\end_layout

\begin_layout Section
Beta Prime
\end_layout

\begin_layout Standard
Defined over 
\begin_inset Formula $0<x<\infty.$
\end_inset

 
\begin_inset Formula $\alpha,\beta>0.$
\end_inset

 (Note the CDF evaluation uses Eq.
 3.194.1 on pg.
 313 of Gradshteyn & Ryzhik (sixth edition).
 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha,\beta\right) & = & \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha-1}\left(1+x\right)^{-\alpha-\beta}\\
F\left(x;\alpha,\beta\right) & = & \frac{\Gamma\left(\alpha+\beta\right)}{\alpha\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha}\,_{2}F_{1}\left(\alpha+\beta,\alpha;1+\alpha;-x\right)\\
G\left(q;\alpha,\beta\right) & = & F^{-1}\left(x;\alpha,\beta\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\left\{ \begin{array}{ccc}
\frac{\Gamma\left(n+\alpha\right)\Gamma\left(\beta-n\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}=\frac{\left(\alpha\right)_{n}}{\left(\beta-n\right)_{n}} &  & \beta>n\\
\infty &  & \textrm{otherwise}\end{array}\right.\]

\end_inset

 Therefore, 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\alpha}{\beta-1}\quad\beta>1\\
\mu_{2} & = & \frac{\alpha\left(\alpha+1\right)}{\left(\beta-2\right)\left(\beta-1\right)}-\frac{\alpha^{2}}{\left(\beta-1\right)^{2}}\quad\beta>2\\
\gamma_{1} & = & \frac{\frac{\alpha\left(\alpha+1\right)\left(\alpha+2\right)}{\left(\beta-3\right)\left(\beta-2\right)\left(\beta-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\quad\beta>3\\
\gamma_{2} & = & \frac{\mu_{4}}{\mu_{2}^{2}}-3\\
\mu_{4} & = & \frac{\alpha\left(\alpha+1\right)\left(\alpha+2\right)\left(\alpha+3\right)}{\left(\beta-4\right)\left(\beta-3\right)\left(\beta-2\right)\left(\beta-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\quad\beta>4\end{eqnarray*}

\end_inset

 
\end_layout

\begin_layout Section
Bradford
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
c & > & 0\\
k & = & \log\left(1+c\right)\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c}{k\left(1+cx\right)}I_{\left(0,1\right)}\left(x\right)\\
F\left(x;c\right) & = & \frac{\log\left(1+cx\right)}{k}\\
G\left(\alpha\; c\right) & = & \frac{\left(1+c\right)^{\alpha}-1}{c}\\
M\left(t\right) & = & \frac{1}{k}e^{-t/c}\left[\textrm{Ei}\left(t+\frac{t}{c}\right)-\textrm{Ei}\left(\frac{t}{c}\right)\right]\\
\mu & = & \frac{c-k}{ck}\\
\mu_{2} & = & \frac{\left(c+2\right)k-2c}{2ck^{2}}\\
\gamma_{1} & = & \frac{\sqrt{2}\left(12c^{2}-9kc\left(c+2\right)+2k^{2}\left(c\left(c+3\right)+3\right)\right)}{\sqrt{c\left(c\left(k-2\right)+2k\right)}\left(3c\left(k-2\right)+6k\right)}\\
\gamma_{2} & = & \frac{c^{3}\left(k-3\right)\left(k\left(3k-16\right)+24\right)+12kc^{2}\left(k-4\right)\left(k-3\right)+6ck^{2}\left(3k-14\right)+12k^{3}}{3c\left(c\left(k-2\right)+2k\right)^{2}}\\
m_{d} & = & 0\\
m_{n} & = & \sqrt{1+c}-1\end{eqnarray*}

\end_inset

 where 
\begin_inset Formula $\textrm{Ei}\left(\textrm{z}\right)$
\end_inset

 is the exponential integral function.
 Also 
\begin_inset Formula \[
h\left[X\right]=\frac{1}{2}\log\left(1+c\right)-\log\left(\frac{c}{\log\left(1+c\right)}\right)\]

\end_inset


\end_layout

\begin_layout Section
Burr
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
c & > & 0\\
d & > & 0\\
k & = & \Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+d\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+d\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c,d\right) & = & \frac{cd}{x^{c+1}\left(1+x^{-c}\right)^{d+1}}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-d}\\
G\left(\alpha;c,d\right) & = & \left(\alpha^{-1/d}-1\right)^{-1/c}\\
\mu & = & \frac{\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+d\right)}{\Gamma\left(d\right)}\\
\mu_{2} & = & \frac{k}{\Gamma^{2}\left(d\right)}\\
\gamma_{1} & = & \frac{1}{\sqrt{k^{3}}}\left[2\Gamma^{3}\left(1-\frac{1}{c}\right)\Gamma^{3}\left(\frac{1}{c}+d\right)+\Gamma^{2}\left(d\right)\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(\frac{3}{c}+d\right)\right.\\
 &  & \left.-3\Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+d\right)\Gamma\left(\frac{2}{c}+d\right)\right]\\
\gamma_{2} & = & -3+\frac{1}{k^{2}}\left[6\Gamma\left(d\right)\Gamma\left(1-\frac{2}{c}\right)\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+d\right)\Gamma\left(\frac{2}{c}+d\right)\right.\\
 &  & -3\Gamma^{4}\left(1-\frac{1}{c}\right)\Gamma^{4}\left(\frac{1}{c}+d\right)+\Gamma^{3}\left(d\right)\Gamma\left(1-\frac{4}{c}\right)\Gamma\left(\frac{4}{c}+d\right)\\
 &  & \left.-4\Gamma^{2}\left(d\right)\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+d\right)\Gamma\left(\frac{3}{c}+d\right)\right]\\
m_{d} & = & \left(\frac{cd-1}{c+1}\right)^{1/c}\,\textrm{if }cd>1\,\textrm{otherwise }0\\
m_{n} & = & \left(2^{1/d}-1\right)^{-1/c}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Cauchy
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\pi\left(1+x^{2}\right)}\\
F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\tan^{-1}x\\
G\left(\alpha\right) & = & \tan\left(\pi\alpha-\frac{\pi}{2}\right)\\
m_{d} & = & 0\\
m_{n} & = & 0\end{eqnarray*}

\end_inset

No finite moments.
 This is the t distribution with one degree of freedom.
 
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(4\pi\right)\\
 & \approx & 2.5310242469692907930.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Chi
\end_layout

\begin_layout Standard
Generated by taking the (positive) square-root of chi-squared variates.
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu\right) & = & \frac{x^{\nu-1}e^{-x^{2}/2}}{2^{\nu/2-1}\Gamma\left(\frac{\nu}{2}\right)}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x;\nu\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x^{2}}{2}\right)\\
G\left(\alpha;\nu\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\alpha\right)}\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\Gamma\left(\frac{v}{2}\right)\,_{1}F_{1}\left(\frac{v}{2};\frac{1}{2};\frac{t^{2}}{2}\right)+\frac{t}{\sqrt{2}}\Gamma\left(\frac{1+\nu}{2}\right)\,_{1}F_{1}\left(\frac{1+\nu}{2};\frac{3}{2};\frac{t^{2}}{2}\right)\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\sqrt{2}\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)}\\
\mu_{2} & = & \nu-\mu^{2}\\
\gamma_{1} & = & \frac{2\mu^{3}+\mu\left(1-2\nu\right)}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{2\nu\left(1-\nu\right)-6\mu^{4}+4\mu^{2}\left(2\nu-1\right)}{\mu_{2}^{2}}\\
m_{d} & = & \sqrt{\nu-1}\quad\nu\geq1\\
m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\frac{1}{2}\right)}\end{eqnarray*}

\end_inset

 
\end_layout

\begin_layout Section
Chi-squared
\end_layout

\begin_layout Standard
This is the gamma distribution with 
\begin_inset Formula $L=0.0$
\end_inset

 and 
\begin_inset Formula $S=2.0$
\end_inset

 and 
\begin_inset Formula $\alpha=\nu/2$
\end_inset

 where 
\begin_inset Formula $\nu$
\end_inset

 is called the degrees of freedom.
 If 
\begin_inset Formula $Z_{1}\ldots Z_{\nu}$
\end_inset

 are all standard normal distributions, then 
\begin_inset Formula $W=\sum_{k}Z_{k}^{2}$
\end_inset

 has (standard) chi-square distribution with 
\begin_inset Formula $\nu$
\end_inset

 degrees of freedom.
 
\end_layout

\begin_layout Standard
The standard form (most often used in standard form only) is 
\begin_inset Formula $x>0$
\end_inset


\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\frac{\nu}{2}\right)}\left(\frac{x}{2}\right)^{\nu/2-1}e^{-x/2}\\
F\left(x;\alpha\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x}{2}\right)\\
G\left(q;\alpha\right) & = & 2\Gamma^{-1}\left(\frac{\nu}{2},q\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1}{2}-t\right)^{\nu/2}}\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \nu\\
\mu_{2} & = & 2\nu\\
\gamma_{1} & = & \frac{2\sqrt{2}}{\sqrt{\nu}}\\
\gamma_{2} & = & \frac{12}{\nu}\\
m_{d} & = & \frac{\nu}{2}-1\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Cosine
\end_layout

\begin_layout Standard
Approximation to the normal distribution.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{2\pi}\left[1+\cos x\right]I_{\left[-\pi,\pi\right]}\left(x\right)\\
F\left(x\right) & = & \frac{1}{2\pi}\left[\pi+x+\sin x\right]I_{\left[-\pi,\pi\right]}\left(x\right)+I_{\left(\pi,\infty\right)}\left(x\right)\\
G\left(\alpha\right) & = & F^{-1}\left(\alpha\right)\\
M\left(t\right) & = & \frac{\sinh\left(\pi t\right)}{\pi t\left(1+t^{2}\right)}\\
\mu=m_{d}=m_{n} & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{3}-2\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{-6\left(\pi^{4}-90\right)}{5\left(\pi^{2}-6\right)^{2}}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(4\pi\right)-1\\
 & \approx & 1.5310242469692907930.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Double Gamma
\end_layout

\begin_layout Standard
The double gamma is the signed version of the Gamma distribution.
 For 
\begin_inset Formula $\alpha>0:$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\alpha\right)}\left|x\right|^{\alpha-1}e^{-\left|x\right|}\\
F\left(x;\alpha\right) & = & \left\{ \begin{array}{ccc}
\frac{1}{2}-\frac{1}{2}\Gamma\left(\alpha,\left|x\right|\right) &  & x\leq0\\
\frac{1}{2}+\frac{1}{2}\Gamma\left(\alpha,\left|x\right|\right) &  & x>0\end{array}\right.\\
G\left(q;\alpha\right) & = & \left\{ \begin{array}{ccc}
-\Gamma^{-1}\left(\alpha,\left|2q-1\right|\right) &  & q\leq\frac{1}{2}\\
\Gamma^{-1}\left(\alpha,\left|2q-1\right|\right) &  & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\right)^{a}}\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu=m_{n} & = & 0\\
\mu_{2} & = & \alpha\left(\alpha+1\right)\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{\left(\alpha+2\right)\left(\alpha+3\right)}{\alpha\left(\alpha+1\right)}-3\\
m_{d} & = & \textrm{NA}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Doubly Non-central F*
\end_layout

\begin_layout Section
Doubly Non-central t*
\end_layout

\begin_layout Section
Double Weibull
\end_layout

\begin_layout Standard
This is a signed form of the Weibull distribution.
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c}{2}\left|x\right|^{c-1}\exp\left(-\left|x\right|^{c}\right)\\
F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
\frac{1}{2}\exp\left(-\left|x\right|^{c}\right) &  & x\leq0\\
1-\frac{1}{2}\exp\left(-\left|x\right|^{c}\right) &  & x>0\end{array}\right.\\
G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
-\log^{1/c}\left(\frac{1}{2q}\right) &  & q\leq\frac{1}{2}\\
\log^{1/c}\left(\frac{1}{2q-1}\right) &  & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}

\end_inset


\begin_inset Formula \[
\mu_{n}^{\prime}=\mu_{n}=\begin{cases}
\Gamma\left(1+\frac{n}{c}\right) & n\textrm{ even}\\
0 & n\textrm{ odd}\end{cases}\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
m_{d}=\mu & = & 0\\
\mu_{2} & = & \Gamma\left(\frac{c+2}{c}\right)\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{\Gamma\left(1+\frac{4}{c}\right)}{\Gamma^{2}\left(1+\frac{2}{c}\right)}\\
m_{d} & = & \textrm{NA bimodal}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Erlang
\end_layout

\begin_layout Standard
This is just the Gamma distribution with shape parameter 
\begin_inset Formula $\alpha=n$
\end_inset

 an integer.
 
\end_layout

\begin_layout Section
Exponential
\end_layout

\begin_layout Standard
This is a special case of the Gamma (and Erlang) distributions with shape
 parameter 
\begin_inset Formula $\left(\alpha=1\right)$
\end_inset

 and the same location and scale parameters.
 The standard form is therefore (
\begin_inset Formula $x\geq0$
\end_inset

) 
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & e^{-x}\\
F\left(x\right) & = & \Gamma\left(1,x\right)=1-e^{-x}\\
G\left(q\right) & = & -\log\left(1-q\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=n!\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{1}{1-t}\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & 1\\
\mu_{2} & = & 1\\
\gamma_{1} & = & 2\\
\gamma_{2} & = & 6\\
m_{d} & = & 0\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
h\left[X\right]=1.\]

\end_inset


\end_layout

\begin_layout Section
Exponentiated Weibull
\end_layout

\begin_layout Standard
Two positive shape parameters 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $c$
\end_inset

 and 
\begin_inset Formula $x\in\left(0,\infty\right)$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,c\right) & = & ac\left[1-\exp\left(-x^{c}\right)\right]^{a-1}\exp\left(-x^{c}\right)x^{c-1}\\
F\left(x;a,c\right) & = & \left[1-\exp\left(-x^{c}\right)\right]^{a}\\
G\left(q;a,c\right) & = & \left[-\log\left(1-q^{1/a}\right)\right]^{1/c}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Exponential Power
\end_layout

\begin_layout Standard
One positive shape parameter 
\begin_inset Formula $b$
\end_inset

.
 Defined for 
\begin_inset Formula $x\geq0.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;b\right) & = & ebx^{b-1}\exp\left[x^{b}-e^{x^{b}}\right]\\
F\left(x;b\right) & = & 1-\exp\left[1-e^{x^{b}}\right]\\
G\left(q;b\right) & = & \log^{1/b}\left[1-\log\left(1-q\right)\right]\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Fatigue Life (Birnbaum-Sanders)
\end_layout

\begin_layout Standard
This distribution's pdf is the average of the inverse-Gaussian 
\begin_inset Formula $\left(\mu=1\right)$
\end_inset

 and reciprocal inverse-Gaussian pdf 
\begin_inset Formula $\left(\mu=1\right)$
\end_inset

.
 We follow the notation of JKB here with 
\begin_inset Formula $\beta=S.$
\end_inset

 for 
\begin_inset Formula $x>0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{x+1}{2c\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2xc^{2}}\right)\\
F\left(x;c\right) & = & \Phi\left(\frac{1}{c}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\right)\\
G\left(q;c\right) & = & \frac{1}{4}\left[c\Phi^{-1}\left(q\right)+\sqrt{c^{2}\left(\Phi^{-1}\left(q\right)\right)^{2}+4}\right]^{2}\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=c\sqrt{2\pi}\exp\left[\frac{1}{c^{2}}\left(1-\sqrt{1-2c^{2}t}\right)\right]\left(1+\frac{1}{\sqrt{1-2c^{2}t}}\right)\]

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{c^{2}}{2}+1\\
\mu_{2} & = & c^{2}\left(\frac{5}{4}c^{2}+1\right)\\
\gamma_{1} & = & \frac{4c\sqrt{11c^{2}+6}}{\left(5c^{2}+4\right)^{3/2}}\\
\gamma_{2} & = & \frac{6c^{2}\left(93c^{2}+41\right)}{\left(5c^{2}+4\right)^{2}}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Fisk (Log Logistic)
\end_layout

\begin_layout Standard
Special case of the Burr distribution with 
\begin_inset Formula $d=1$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
c & > & 0\\
k & = & \Gamma\left(1-\frac{2}{c}\right)\Gamma\left(\frac{2}{c}+1\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c,d\right) & = & \frac{cx^{c-1}}{\left(1+x^{c}\right)^{2}}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x;c,d\right) & = & \left(1+x^{-c}\right)^{-1}\\
G\left(\alpha;c,d\right) & = & \left(\alpha^{-1}-1\right)^{-1/c}\\
\mu & = & \Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\\
\mu_{2} & = & k\\
\gamma_{1} & = & \frac{1}{\sqrt{k^{3}}}\left[2\Gamma^{3}\left(1-\frac{1}{c}\right)\Gamma^{3}\left(\frac{1}{c}+1\right)+\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(\frac{3}{c}+1\right)\right.\\
 &  & \left.-3\Gamma\left(1-\frac{2}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\Gamma\left(\frac{2}{c}+1\right)\right]\\
\gamma_{2} & = & -3+\frac{1}{k^{2}}\left[6\Gamma\left(1-\frac{2}{c}\right)\Gamma^{2}\left(1-\frac{1}{c}\right)\Gamma^{2}\left(\frac{1}{c}+1\right)\Gamma\left(\frac{2}{c}+1\right)\right.\\
 &  & -3\Gamma^{4}\left(1-\frac{1}{c}\right)\Gamma^{4}\left(\frac{1}{c}+1\right)+\Gamma\left(1-\frac{4}{c}\right)\Gamma\left(\frac{4}{c}+1\right)\\
 &  & \left.-4\Gamma\left(1-\frac{3}{c}\right)\Gamma\left(1-\frac{1}{c}\right)\Gamma\left(\frac{1}{c}+1\right)\Gamma\left(\frac{3}{c}+1\right)\right]\\
m_{d} & = & \left(\frac{c-1}{c+1}\right)^{1/c}\,\textrm{if }c>1\,\textrm{otherwise }0\\
m_{n} & = & 1\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=2-\log c.\]

\end_inset


\end_layout

\begin_layout Section
Folded Cauchy
\end_layout

\begin_layout Standard
This formula can be expressed in terms of the standard formulas for the
 Cauchy distribution (call the cdf 
\begin_inset Formula $C\left(x\right)$
\end_inset

 and the pdf 
\begin_inset Formula $d\left(x\right)$
\end_inset

).
 if 
\begin_inset Formula $Y$
\end_inset

 is cauchy then 
\begin_inset Formula $\left|Y\right|$
\end_inset

 is folded cauchy.
 Note that 
\begin_inset Formula $x\geq0.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{1}{\pi\left(1+\left(x-c\right)^{2}\right)}+\frac{1}{\pi\left(1+\left(x+c\right)^{2}\right)}\\
F\left(x;c\right) & = & \frac{1}{\pi}\tan^{-1}\left(x-c\right)+\frac{1}{\pi}\tan^{-1}\left(x+c\right)\\
G\left(q;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
No moments
\end_layout

\begin_layout Section
Folded Normal
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $Z$
\end_inset

 is Normal with mean 
\begin_inset Formula $L$
\end_inset

 and 
\begin_inset Formula $\sigma=S$
\end_inset

, then 
\begin_inset Formula $\left|Z\right|$
\end_inset

 is a folded normal with shape parameter 
\begin_inset Formula $c=\left|L\right|/S$
\end_inset

, location parameter 
\begin_inset Formula $0$
\end_inset

 and scale parameter 
\begin_inset Formula $S$
\end_inset

.
 This is a special case of the non-central chi distribution with one-degree
 of freedom and non-centrality parameter 
\begin_inset Formula $c^{2}.$
\end_inset

 Note that 
\begin_inset Formula $c\geq0$
\end_inset

.
 The standard form of the folded normal is 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \sqrt{\frac{2}{\pi}}\cosh\left(cx\right)\exp\left(-\frac{x^{2}+c^{2}}{2}\right)\\
F\left(x;c\right) & = & \Phi\left(x-c\right)-\Phi\left(-x-c\right)=\Phi\left(x-c\right)+\Phi\left(x+c\right)-1\\
G\left(\alpha;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
M\left(t\right)=\exp\left[\frac{t}{2}\left(t-2c\right)\right]\left(1+e^{2ct}\right)\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
k & = & \textrm{erf}\left(\frac{c}{\sqrt{2}}\right)\\
p & = & \exp\left(-\frac{c^{2}}{2}\right)\\
\mu & = & \sqrt{\frac{2}{\pi}}p+ck\\
\mu_{2} & = & c^{2}+1-\mu^{2}\\
\gamma_{1} & = & \frac{\sqrt{\frac{2}{\pi}}p^{3}\left(4-\frac{\pi}{p^{2}}\left(2c^{2}+1\right)\right)+2ck\left(6p^{2}+3cpk\sqrt{2\pi}+\pi c\left(k^{2}-1\right)\right)}{\pi\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{c^{4}+6c^{2}+3+6\left(c^{2}+1\right)\mu^{2}-3\mu^{4}-4p\mu\left(\sqrt{\frac{2}{\pi}}\left(c^{2}+2\right)+\frac{ck}{p}\left(c^{2}+3\right)\right)}{\mu_{2}^{2}}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Fratio (or F)
\end_layout

\begin_layout Standard
Defined for 
\begin_inset Formula $x>0$
\end_inset

.
 The distribution of 
\begin_inset Formula $\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)$
\end_inset

 if 
\begin_inset Formula $X_{1}$
\end_inset

 is chi-squared with 
\begin_inset Formula $v_{1}$
\end_inset

 degrees of freedom and 
\begin_inset Formula $X_{2}$
\end_inset

 is chi-squared with 
\begin_inset Formula $v_{2}$
\end_inset

 degrees of freedom.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu_{1},\nu_{2}\right) & = & \frac{\nu_{2}^{\nu_{2}/2}\nu_{1}^{\nu_{1}/2}x^{\nu_{1}/2-1}}{\left(\nu_{2}+\nu_{1}x\right)^{\left(\nu_{1}+\nu_{2}\right)/2}B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)}\\
F\left(x;v_{1},v_{2}\right) & = & I\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2},\frac{\nu_{2}x}{\nu_{2}+\nu_{1}x}\right)\\
G\left(q;\nu_{1},\nu_{2}\right) & = & \left[\frac{\nu_{2}}{I^{-1}\left(\nu_{1}/2,\nu_{2}/2,q\right)}-\frac{\nu_{1}}{\nu_{2}}\right]^{-1}.\end{eqnarray*}

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\nu_{2}}{\nu_{2}-2}\quad\nu_{2}>2\\
\mu_{2} & = & \frac{2\nu_{2}^{2}\left(\nu_{1}+\nu_{2}-2\right)}{\nu_{1}\left(\nu_{2}-2\right)^{2}\left(\nu_{2}-4\right)}\quad v_{2}>4\\
\gamma_{1} & = & \frac{2\left(2\nu_{1}+\nu_{2}-2\right)}{\nu_{2}-6}\sqrt{\frac{2\left(\nu_{2}-4\right)}{\nu_{1}\left(\nu_{1}+\nu_{2}-2\right)}}\quad\nu_{2}>6\\
\gamma_{2} & = & \frac{3\left[8+\left(\nu_{2}-6\right)\gamma_{1}^{2}\right]}{2\nu-16}\quad\nu_{2}>8\end{eqnarray*}

\end_inset

 
\end_layout

\begin_layout Section
Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)
\end_layout

\begin_layout Standard
A type of extreme-value distribution with a lower bound.
 Defined for 
\begin_inset Formula $x>0$
\end_inset

 and 
\begin_inset Formula $c>0$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & cx^{c-1}\exp\left(-x^{c}\right)\\
F\left(x;c\right) & = & 1-\exp\left(-x^{c}\right)\\
G\left(q;c\right) & = & \left[-\log\left(1-q\right)\right]^{1/c}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
\mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right)\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \Gamma\left(1+\frac{1}{c}\right)\\
\mu_{2} & = & \Gamma\left(1+\frac{2}{c}\right)-\Gamma^{2}\left(1-\frac{1}{c}\right)\\
\gamma_{1} & = & \frac{\Gamma\left(1+\frac{3}{c}\right)-3\Gamma\left(1+\frac{2}{c}\right)\Gamma\left(1+\frac{1}{c}\right)+2\Gamma^{3}\left(1+\frac{1}{c}\right)}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\Gamma\left(1+\frac{4}{c}\right)-4\Gamma\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{3}{c}\right)+6\Gamma^{2}\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{2}{c}\right)-\Gamma^{4}\left(1+\frac{1}{c}\right)}{\mu_{2}^{2}}-3\\
m_{d} & = & \left(\frac{c}{1+c}\right)^{1/c}\\
m_{n} & = & G\left(\frac{1}{2};c\right)\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]

\end_inset

 where 
\begin_inset Formula $\gamma$
\end_inset

 is Euler's constant and equal to 
\begin_inset Formula \[
\gamma\approx0.57721566490153286061.\]

\end_inset

 
\end_layout

\begin_layout Section
Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)
\end_layout

\begin_layout Standard
Defined for 
\begin_inset Formula $x<0$
\end_inset

 and 
\begin_inset Formula $c>0$
\end_inset

.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & c\left(-x\right)^{c-1}\exp\left(-\left(-x\right)^{c}\right)\\
F\left(x;c\right) & = & \exp\left(-\left(-x\right)^{c}\right)\\
G\left(q;c\right) & = & -\left(-\log q\right)^{1/c}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
The mean is the negative of the right-skewed Frechet distribution given
 above, and the other statistical parameters can be computed from
\end_layout

\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right).\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]

\end_inset

 where 
\begin_inset Formula $\gamma$
\end_inset

 is Euler's constant and equal to 
\begin_inset Formula \[
\gamma\approx0.57721566490153286061.\]

\end_inset


\end_layout

\begin_layout Section
Gamma
\end_layout

\begin_layout Standard
The standard form for the gamma distribution is 
\begin_inset Formula $\left(\alpha>0\right)$
\end_inset

 valid for 
\begin_inset Formula $x\geq0$
\end_inset

.
\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha\right) & = & \frac{1}{\Gamma\left(\alpha\right)}x^{\alpha-1}e^{-x}\\
F\left(x;\alpha\right) & = & \Gamma\left(\alpha,x\right)\\
G\left(q;\alpha\right) & = & \Gamma^{-1}\left(\alpha,q\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \alpha\\
\mu_{2} & = & \alpha\\
\gamma_{1} & = & \frac{2}{\sqrt{\alpha}}\\
\gamma_{2} & = & \frac{6}{\alpha}\\
m_{d} & = & \alpha-1\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(a\right)\]

\end_inset

 where 
\begin_inset Formula \[
\Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.\]

\end_inset


\end_layout

\begin_layout Section
Generalized Logistic
\end_layout

\begin_layout Standard
Has been used in the analysis of extreme values.
 Has one shape parameter 
\begin_inset Formula $c>0.$
\end_inset

 And 
\begin_inset Formula $x>0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{c+1}}\\
F\left(x;c\right) & = & \frac{1}{\left[1+\exp\left(-x\right)\right]^{c}}\\
G\left(q;c\right) & = & -\log\left(q^{-1/c}-1\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\frac{c}{1-t}\,_{2}F_{1}\left(1+c,\,1-t\,;\,2-t\,;-1\right)\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \gamma+\psi_{0}\left(c\right)\\
\mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\\
\gamma_{1} & = & \frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\mu_{2}^{2}}\\
m_{d} & = & \log c\\
m_{n} & = & -\log\left(2^{1/c}-1\right)\end{eqnarray*}

\end_inset

 Note that the polygamma function is 
\begin_inset Formula \begin{eqnarray*}
\psi_{n}\left(z\right) & = & \frac{d^{n+1}}{dz^{n+1}}\log\Gamma\left(z\right)\\
 & = & \left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\
 & = & \left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\end{eqnarray*}

\end_inset

 where 
\begin_inset Formula $\zeta\left(k,x\right)$
\end_inset

 is a generalization of the Riemann zeta function called the Hurwitz zeta
 function Note that 
\begin_inset Formula $\zeta\left(n\right)\equiv\zeta\left(n,1\right)$
\end_inset


\end_layout

\begin_layout Section
Generalized Pareto
\end_layout

\begin_layout Standard
Shape parameter 
\begin_inset Formula $c\neq0$
\end_inset

 and defined for 
\begin_inset Formula $x\geq0$
\end_inset

 for all 
\begin_inset Formula $c$
\end_inset

 and 
\begin_inset Formula $x<\frac{1}{\left|c\right|}$
\end_inset

 if 
\begin_inset Formula $c$
\end_inset

 is negative.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \left(1+cx\right)^{-1-\frac{1}{c}}\\
F\left(x;c\right) & = & 1-\frac{1}{\left(1+cx\right)^{1/c}}\\
G\left(q;c\right) & = & \frac{1}{c}\left[\left(\frac{1}{1-q}\right)^{c}-1\right]\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=\left\{ \begin{array}{cc}
\left(-\frac{t}{c}\right)^{\frac{1}{c}}e^{-\frac{t}{c}}\left[\Gamma\left(1-\frac{1}{c}\right)+\Gamma\left(-\frac{1}{c},-\frac{t}{c}\right)-\pi\csc\left(\frac{\pi}{c}\right)/\Gamma\left(\frac{1}{c}\right)\right] & c>0\\
\left(\frac{\left|c\right|}{t}\right)^{1/\left|c\right|}\Gamma\left[\frac{1}{\left|c\right|},\frac{t}{\left|c\right|}\right] & c<0\end{array}\right.\]

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\left(-1\right)^{n}}{c^{n}}\sum_{k=0}^{n}\left(\begin{array}{c}
n\\
k\end{array}\right)\frac{\left(-1\right)^{k}}{1-ck}\quad cn<1\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu_{1}^{\prime} & = & \frac{1}{1-c}\quad c<1\\
\mu_{2}^{\prime} & = & \frac{2}{\left(1-2c\right)\left(1-c\right)}\quad c<\frac{1}{2}\\
\mu_{3}^{\prime} & = & \frac{6}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)}\quad c<\frac{1}{3}\\
\mu_{4}^{\prime} & = & \frac{24}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)\left(1-4c\right)}\quad c<\frac{1}{4}\end{eqnarray*}

\end_inset

 Thus,
\begin_inset Formula \begin{eqnarray*}
\mu & = & \mu_{1}^{\prime}\\
\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=1+c\quad c>0\]

\end_inset


\end_layout

\begin_layout Section
Generalized Exponential
\end_layout

\begin_layout Standard
Three positive shape parameters for 
\begin_inset Formula $x\geq0.$
\end_inset

 Note that 
\begin_inset Formula $a,b,$
\end_inset

 and 
\begin_inset Formula $c$
\end_inset

 are all 
\begin_inset Formula $>0.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,b,c\right) & = & \left(a+b\left(1-e^{-cx}\right)\right)\exp\left[ax-bx+\frac{b}{c}\left(1-e^{-cx}\right)\right]\\
F\left(x;a,b,c\right) & = & 1-\exp\left[ax-bx+\frac{b}{c}\left(1-e^{-cx}\right)\right]\\
G\left(q;a,b,c\right) & = & F^{-1}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Generalized Extreme Value
\end_layout

\begin_layout Standard
Extreme value distributions with shape parameter 
\begin_inset Formula $c$
\end_inset

.
 
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $c>0$
\end_inset

 defined on 
\begin_inset Formula $-\infty<x\leq1/c.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \exp\left[-\left(1-cx\right)^{1/c}\right]\left(1-cx\right)^{1/c-1}\\
F\left(x;c\right) & = & \exp\left[-\left(1-cx\right)^{1/c}\right]\\
G\left(q;c\right) & = & \frac{1}{c}\left[1-\left(-\log q\right)^{c}\right]\end{eqnarray*}

\end_inset


\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{1}{c^{n}}\sum_{k=0}^{n}\left(\begin{array}{c}
n\\
k\end{array}\right)\left(-1\right)^{k}\Gamma\left(ck+1\right)\quad cn>-1\]

\end_inset

 So,
\begin_inset Formula \begin{eqnarray*}
\mu_{1}^{\prime} & = & \frac{1}{c}\left(1-\Gamma\left(1+c\right)\right)\quad c>-1\\
\mu_{2}^{\prime} & = & \frac{1}{c^{2}}\left(1-2\Gamma\left(1+c\right)+\Gamma\left(1+2c\right)\right)\quad c>-\frac{1}{2}\\
\mu_{3}^{\prime} & = & \frac{1}{c^{3}}\left(1-3\Gamma\left(1+c\right)+3\Gamma\left(1+2c\right)-\Gamma\left(1+3c\right)\right)\quad c>-\frac{1}{3}\\
\mu_{4}^{\prime} & = & \frac{1}{c^{4}}\left(1-4\Gamma\left(1+c\right)+6\Gamma\left(1+2c\right)-4\Gamma\left(1+3c\right)+\Gamma\left(1+4c\right)\right)\quad c>-\frac{1}{4}\end{eqnarray*}

\end_inset

 For 
\begin_inset Formula $c<0$
\end_inset

 defined on 
\begin_inset Formula $\frac{1}{c}\leq x<\infty.$
\end_inset

 For 
\begin_inset Formula $c=0$
\end_inset

 defined over all space 
\begin_inset Formula \begin{eqnarray*}
f\left(x;0\right) & = & \exp\left[-e^{-x}\right]e^{-x}\\
F\left(x;0\right) & = & \exp\left[-e^{-x}\right]\\
G\left(q;0\right) & = & -\log\left(-\log q\right)\end{eqnarray*}

\end_inset

 This is just the (left-skewed) Gumbel distribution for c=0.
 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \gamma=-\psi_{0}\left(1\right)\\
\mu_{2} & = & \frac{\pi^{2}}{6}\\
\gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
\gamma_{2} & = & \frac{12}{5}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Generalized Gamma
\end_layout

\begin_layout Standard
A general probability form that reduces to many common distributions: 
\begin_inset Formula $x>0$
\end_inset

 
\begin_inset Formula $a>0$
\end_inset

 and 
\begin_inset Formula $c\neq0.$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,c\right) & = & \frac{\left|c\right|x^{ca-1}}{\Gamma\left(a\right)}\exp\left(-x^{c}\right)\\
F\left(x;a,c\right) & = & \begin{array}{cc}
\frac{\Gamma\left(a,x^{c}\right)}{\Gamma\left(a\right)} & c>0\\
1-\frac{\Gamma\left(a,x^{c}\right)}{\Gamma\left(a\right)} & c<0\end{array}\\
G\left(q;a,c\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{1/c}\quad c>0\\
 &  & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)\left(1-q\right)\right]\right\} ^{1/c}\quad c<0\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)}\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\Gamma\left(a+\frac{1}{c}\right)}{\Gamma\left(a\right)}\\
\mu_{2} & = & \frac{\Gamma\left(a+\frac{2}{c}\right)}{\Gamma\left(a\right)}-\mu^{2}\\
\gamma_{1} & = & \frac{\Gamma\left(a+\frac{3}{c}\right)/\Gamma\left(a\right)-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\Gamma\left(a+\frac{4}{c}\right)/\Gamma\left(a\right)-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\\
m_{d} & = & \left(\frac{ac-1}{c}\right)^{1/c}.\end{eqnarray*}

\end_inset

 Special cases are Weibull 
\begin_inset Formula $\left(a=1\right)$
\end_inset

, half-normal 
\begin_inset Formula $\left(a=1/2,c=2\right)$
\end_inset

 and ordinary gamma distributions 
\begin_inset Formula $c=1.$
\end_inset

 If 
\begin_inset Formula $c=-1$
\end_inset

 then it is the inverted gamma distribution.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\log\Gamma\left(a\right)-\log\left|c\right|.\]

\end_inset


\end_layout

\begin_layout Section
Generalized Half-Logistic
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $x\in\left[0,1/c\right]$
\end_inset

 and 
\begin_inset Formula $c>0$
\end_inset

 we have 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{2\left(1-cx\right)^{\frac{1}{c}-1}}{\left(1+\left(1-cx\right)^{1/c}\right)^{2}}\\
F\left(x;c\right) & = & \frac{1-\left(1-cx\right)^{1/c}}{1+\left(1-cx\right)^{1/c}}\\
G\left(q;c\right) & = & \frac{1}{c}\left[1-\left(\frac{1-q}{1+q}\right)^{c}\right]\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & 2-\left(2c+1\right)\log2.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Gilbrat
\end_layout

\begin_layout Standard
Special case of the log-normal with 
\begin_inset Formula $\sigma=1$
\end_inset

 and 
\begin_inset Formula $S=1.0$
\end_inset

 (typically also 
\begin_inset Formula $L=0.0$
\end_inset

) 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\sigma\right) & = & \frac{1}{x\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\log x\right)^{2}\right]\\
F\left(x;\sigma\right) & = & \Phi\left(\log x\right)=\frac{1}{2}\left[1+\textrm{erf}\left(\frac{\log x}{\sqrt{2}}\right)\right]\\
G\left(q;\sigma\right) & = & \exp\left\{ \Phi^{-1}\left(q\right)\right\} \end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \sqrt{e}\\
\mu_{2} & = & e\left[e-1\right]\\
\gamma_{1} & = & \sqrt{e-1}\left(2+e\right)\\
\gamma_{2} & = & e^{4}+2e^{3}+3e^{2}-6\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\
 & \approx & 1.4189385332046727418\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Gompertz (Truncated Gumbel)
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $x\geq0$
\end_inset

 and 
\begin_inset Formula $c>0$
\end_inset

.
 In JKB the two shape parameters 
\begin_inset Formula $b,a$
\end_inset

 are reduced to the single shape-parameter 
\begin_inset Formula $c=b/a$
\end_inset

.
 As 
\begin_inset Formula $a$
\end_inset

 is just a scale parameter when 
\begin_inset Formula $a\neq0$
\end_inset

.
 If 
\begin_inset Formula $a=0,$
\end_inset

 the distribution reduces to the exponential distribution scaled by 
\begin_inset Formula $1/b.$
\end_inset

 Thus, the standard form is given as 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & ce^{x}\exp\left[-c\left(e^{x}-1\right)\right]\\
F\left(x;c\right) & = & 1-\exp\left[-c\left(e^{x}-1\right)\right]\\
G\left(q;c\right) & = & \log\left[1-\frac{1}{c}\log\left(1-q\right)\right]\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=1-\log\left(c\right)-e^{c}\textrm{Ei}\left(1,c\right),\]

\end_inset

where 
\begin_inset Formula \[
\textrm{Ei}\left(n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\right)dt\]

\end_inset


\end_layout

\begin_layout Section
Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)
\end_layout

\begin_layout Standard
One of a clase of extreme value distributions (right-skewed).
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \exp\left(-\left(x+e^{-x}\right)\right)\\
F\left(x\right) & = & \exp\left(-e^{-x}\right)\\
G\left(q\right) & = & -\log\left(-\log\left(q\right)\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\Gamma\left(1-t\right)\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \gamma=-\psi_{0}\left(1\right)\\
\mu_{2} & = & \frac{\pi^{2}}{6}\\
\gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
\gamma_{2} & = & \frac{12}{5}\\
m_{d} & = & 0\\
m_{n} & = & -\log\left(\log2\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]\approx1.0608407169541684911\]

\end_inset


\end_layout

\begin_layout Section
Gumbel Left-skewed (for minimum order statistic)
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \exp\left(x-e^{x}\right)\\
F\left(x\right) & = & 1-\exp\left(-e^{x}\right)\\
G\left(q\right) & = & \log\left(-\log\left(1-q\right)\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\Gamma\left(1+t\right)\]

\end_inset

 Note, that 
\begin_inset Formula $\mu$
\end_inset

 is negative the mean for the right-skewed distribution.
 Similar for median and mode.
 All other moments are the same.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]\approx1.0608407169541684911.\]

\end_inset


\end_layout

\begin_layout Section
HalfCauchy
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $Z$
\end_inset

 is Hyperbolic Secant distributed then 
\begin_inset Formula $e^{Z}$
\end_inset

 is Half-Cauchy distributed.
 Also, if 
\begin_inset Formula $W$
\end_inset

 is (standard) Cauchy distributed, then 
\begin_inset Formula $\left|W\right|$
\end_inset

 is Half-Cauchy distributed.
 Special case of the Folded Cauchy distribution with 
\begin_inset Formula $c=0.$
\end_inset

 The standard form is 
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{2}{\pi\left(1+x^{2}\right)}I_{[0,\infty)}\left(x\right)\\
F\left(x\right) & = & \frac{2}{\pi}\arctan\left(x\right)I_{\left[0,\infty\right]}\left(x\right)\\
G\left(q\right) & = & \tan\left(\frac{\pi}{2}q\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\cos t+\frac{2}{\pi}\left[\textrm{Si}\left(t\right)\cos t-\textrm{Ci}\left(\textrm{-}t\right)\sin t\right]\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
m_{d} & = & 0\\
m_{n} & = & \tan\left(\frac{\pi}{4}\right)\end{eqnarray*}

\end_inset

 No moments, as the integrals diverge.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(2\pi\right)\\
 & \approx & 1.8378770664093454836.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
HalfNormal
\end_layout

\begin_layout Standard
This is a special case of the chi distribution with 
\begin_inset Formula $L=a$
\end_inset

 and 
\begin_inset Formula $S=b$
\end_inset

 and 
\begin_inset Formula $\nu=1.$
\end_inset

 This is also a special case of the folded normal with shape parameter 
\begin_inset Formula $c=0$
\end_inset

 and 
\begin_inset Formula $S=S.$
\end_inset

 If 
\begin_inset Formula $Z$
\end_inset

 is (standard) normally distributed then, 
\begin_inset Formula $\left|Z\right|$
\end_inset

 is half-normal.
 The standard form is 
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \sqrt{\frac{2}{\pi}}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x\right) & = & 2\Phi\left(x\right)-1\\
G\left(q\right) & = & \Phi^{-1}\left(\frac{1+q}{2}\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
M\left(t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right)\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \sqrt{\frac{2}{\pi}}\\
\mu_{2} & = & 1-\frac{2}{\pi}\\
\gamma_{1} & = & \frac{\sqrt{2}\left(4-\pi\right)}{\left(\pi-2\right)^{3/2}}\\
\gamma_{2} & = & \frac{8\left(\pi-3\right)}{\left(\pi-2\right)^{2}}\\
m_{d} & = & 0\\
m_{n} & = & \Phi^{-1}\left(\frac{3}{4}\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(\sqrt{\frac{\pi e}{2}}\right)\\
 & \approx & 0.72579135264472743239.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Half-Logistic 
\end_layout

\begin_layout Standard
In the limit as 
\begin_inset Formula $c\rightarrow\infty$
\end_inset

 for the generalized half-logistic we have the half-logistic defined over
 
\begin_inset Formula $x\geq0.$
\end_inset

 Also, the distribution of 
\begin_inset Formula $\left|X\right|$
\end_inset

 where 
\begin_inset Formula $X$
\end_inset

 has logistic distribtution.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{2e^{-x}}{\left(1+e^{-x}\right)^{2}}=\frac{1}{2}\textrm{sech}^{2}\left(\frac{x}{2}\right)\\
F\left(x\right) & = & \frac{1-e^{-x}}{1+e^{-x}}=\tanh\left(\frac{x}{2}\right)\\
G\left(q\right) & = & \log\left(\frac{1+q}{1-q}\right)=2\textrm{arctanh}\left(q\right)\end{eqnarray*}

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
M\left(t\right)=1-t\psi_{0}\left(\frac{1}{2}-\frac{t}{2}\right)+t\psi_{0}\left(1-\frac{t}{2}\right)\]

\end_inset


\begin_inset Formula \[
\mu_{n}^{\prime}=2\left(1-2^{1-n}\right)n!\zeta\left(n\right)\quad n\neq1\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu_{1}^{\prime} & = & 2\log\left(2\right)\\
\mu_{2}^{\prime} & = & 2\zeta\left(2\right)=\frac{\pi^{2}}{3}\\
\mu_{3}^{\prime} & = & 9\zeta\left(3\right)\\
\mu_{4}^{\prime} & = & 42\zeta\left(4\right)=\frac{7\pi^{4}}{15}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & 2-\log\left(2\right)\\
 & \approx & 1.3068528194400546906.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Hyperbolic Secant
\end_layout

\begin_layout Standard
Related to the logistic distribution and used in lifetime analysis.
 Standard form is (defined over all 
\begin_inset Formula $x$
\end_inset

)
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\pi}\textrm{sech}\left(x\right)\\
F\left(x\right) & = & \frac{2}{\pi}\arctan\left(e^{x}\right)\\
G\left(q\right) & = & \log\left(\tan\left(\frac{\pi}{2}q\right)\right)\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
M\left(t\right)=\sec\left(\frac{\pi}{2}t\right)\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu_{n}^{\prime} & = & \frac{1+\left(-1\right)^{n}}{2\pi2^{2n}}n!\left[\zeta\left(n+1,\frac{1}{4}\right)-\zeta\left(n+1,\frac{3}{4}\right)\right]\\
 & = & \left\{ \begin{array}{cc}
0 & n\textrm{ odd}\\
C_{n/2}\frac{\pi^{n}}{2^{n}} & n\textrm{ even}\end{array}\right.\end{eqnarray*}

\end_inset

 where 
\begin_inset Formula $C_{m}$
\end_inset

 is an integer given by 
\begin_inset Formula \begin{eqnarray*}
C_{m} & = & \frac{\left(2m\right)!\left[\zeta\left(2m+1,\frac{1}{4}\right)-\zeta\left(2m+1,\frac{3}{4}\right)\right]}{\pi^{2m+1}2^{2m}}\\
 & = & 4\left(-1\right)^{m-1}\frac{16^{m}}{2m+1}B_{2m+1}\left(\frac{1}{4}\right)\end{eqnarray*}

\end_inset

where 
\begin_inset Formula $B_{2m+1}\left(\frac{1}{4}\right)$
\end_inset

 is the Bernoulli polynomial of order 
\begin_inset Formula $2m+1$
\end_inset

 evaluated at 
\begin_inset Formula $1/4.$
\end_inset

 Thus 
\begin_inset Formula \[
\mu_{n}^{\prime}=\left\{ \begin{array}{cc}
0 & n\textrm{ odd}\\
4\left(-1\right)^{n/2-1}\frac{\left(2\pi\right)^{n}}{n+1}B_{n+1}\left(\frac{1}{4}\right) & n\textrm{ even}\end{array}\right.\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{4}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & 2\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\log\left(2\pi\right).\]

\end_inset


\end_layout

\begin_layout Section
Gauss Hypergeometric 
\end_layout

\begin_layout Standard
\begin_inset Formula $x\in\left[0,1\right]$
\end_inset

, 
\begin_inset Formula $\alpha>0,\,\beta>0$
\end_inset


\begin_inset Formula \[
C^{-1}=B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
f\left(x;\alpha,\beta,\gamma,z\right) & = & Cx^{\alpha-1}\frac{\left(1-x\right)^{\beta-1}}{\left(1+zx\right)^{\gamma}}\\
\mu_{n}^{\prime} & = & \frac{B\left(n+\alpha,\beta\right)}{B\left(\alpha,\beta\right)}\frac{\,_{2}F_{1}\left(\gamma,\alpha+n;\alpha+\beta+n;-z\right)}{\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Inverted Gamma
\end_layout

\begin_layout Standard
Special case of the generalized Gamma distribution with 
\begin_inset Formula $c=-1$
\end_inset

 and 
\begin_inset Formula $a>0$
\end_inset

, 
\begin_inset Formula $x>0$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a\right) & = & \frac{x^{-a-1}}{\Gamma\left(a\right)}\exp\left(-\frac{1}{x}\right)\\
F\left(x;a\right) & = & \frac{\Gamma\left(a,\frac{1}{x}\right)}{\Gamma\left(a\right)}\\
G\left(q;a\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{-1}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{1}{a-1}\quad a>1\\
\mu_{2} & = & \frac{1}{\left(a-2\right)\left(a-1\right)}-\mu^{2}\quad a>2\\
\gamma_{1} & = & \frac{\frac{1}{\left(a-3\right)\left(a-2\right)\left(a-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\frac{1}{\left(a-4\right)\left(a-3\right)\left(a-2\right)\left(a-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}

\end_inset


\begin_inset Formula \[
m_{d}=\frac{1}{a+1}\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=a-\left(a+1\right)\Psi\left(a\right)+\log\Gamma\left(a\right).\]

\end_inset


\end_layout

\begin_layout Section
Inverse Normal (Inverse Gaussian)
\end_layout

\begin_layout Standard
The standard form involves the shape parameter 
\begin_inset Formula $\mu$
\end_inset

 (in most definitions, 
\begin_inset Formula $L=0.0$
\end_inset

 is used).
 (In terms of the regress documentation 
\begin_inset Formula $\mu=A/B$
\end_inset

) and 
\begin_inset Formula $B=S$
\end_inset

 and 
\begin_inset Formula $L$
\end_inset

 is not a parameter in that distribution.
 A standard form is 
\begin_inset Formula $x>0$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2x\mu^{2}}\right).\\
F\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\\
G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \mu\\
\mu_{2} & = & \mu^{3}\\
\gamma_{1} & = & 3\sqrt{\mu}\\
\gamma_{2} & = & 15\mu\\
m_{d} & = & \frac{\mu}{2}\left(\sqrt{9\mu^{2}+4}-3\mu\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
This is related to the canonical form or JKB 
\begin_inset Quotes eld
\end_inset

two-parameter
\begin_inset Quotes erd
\end_inset

 inverse Gaussian when written in it's full form with scale parameter 
\begin_inset Formula $S$
\end_inset

 and location parameter 
\begin_inset Formula $L$
\end_inset

 by taking 
\begin_inset Formula $L=0$
\end_inset

 and 
\begin_inset Formula $S\equiv\lambda,$
\end_inset

 then 
\begin_inset Formula $\mu S$
\end_inset

 is equal to 
\begin_inset Formula $\mu_{2}$
\end_inset

 where 
\begin_inset Formula $\mu_{2}$
\end_inset

 is the parameter used by JKB.
 We prefer this form because of it's consistent use of the scale parameter.
 Notice that in JKB the skew 
\begin_inset Formula $\left(\sqrt{\beta_{1}}\right)$
\end_inset

 and the kurtosis (
\begin_inset Formula $\beta_{2}-3$
\end_inset

) are both functions only of 
\begin_inset Formula $\mu_{2}/\lambda=\mu S/S=\mu$
\end_inset

 as shown here, while the variance and mean of the standard form here are
 transformed appropriately.
 
\end_layout

\begin_layout Section
Inverted Weibull
\end_layout

\begin_layout Standard
Shape parameter 
\begin_inset Formula $c>0$
\end_inset

 and 
\begin_inset Formula $x>0$
\end_inset

.
 Then 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & cx^{-c-1}\exp\left(-x^{-c}\right)\\
F\left(x;c\right) & = & \exp\left(-x^{-c}\right)\\
G\left(q;c\right) & = & \left(-\log q\right)^{-1/c}\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=1+\gamma+\frac{\gamma}{c}-\log\left(c\right)\]

\end_inset

 where 
\begin_inset Formula $\gamma$
\end_inset

 is Euler's constant.
\end_layout

\begin_layout Section
Johnson SB
\end_layout

\begin_layout Standard
Defined for 
\begin_inset Formula $x\in\left(0,1\right)$
\end_inset

 with two shape parameters 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b>0.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,b\right) & = & \frac{b}{x\left(1-x\right)}\phi\left(a+b\log\frac{x}{1-x}\right)\\
F\left(x;a,b\right) & = & \Phi\left(a+b\log\frac{x}{1-x}\right)\\
G\left(q;a,b\right) & = & \frac{1}{1+\exp\left[-\frac{1}{b}\left(\Phi^{-1}\left(q\right)-a\right)\right]}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Johnson SU
\end_layout

\begin_layout Standard
Defined for all 
\begin_inset Formula $x$
\end_inset

 with two shape parameters 
\begin_inset Formula $a$
\end_inset

 and 
\begin_inset Formula $b>0$
\end_inset

.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,b\right) & = & \frac{b}{\sqrt{x^{2}+1}}\phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\
F\left(x;a,b\right) & = & \Phi\left(a+b\log\left(x+\sqrt{x^{2}+1}\right)\right)\\
G\left(q;a,b\right) & = & \sinh\left[\frac{\Phi^{-1}\left(q\right)-a}{b}\right]\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
KSone
\end_layout

\begin_layout Section
KStwo
\end_layout

\begin_layout Section
Laplace (Double Exponential, Bilateral Expoooonential)
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{2}e^{-\left|x\right|}\\
F\left(x\right) & = & \left\{ \begin{array}{ccc}
\frac{1}{2}e^{x} &  & x\leq0\\
1-\frac{1}{2}e^{-x} &  & x>0\end{array}\right.\\
G\left(q\right) & = & \left\{ \begin{array}{ccc}
\log\left(2q\right) &  & q\leq\frac{1}{2}\\
-\log\left(2-2q\right) &  & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & 2\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & 3\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
The ML estimator of the location parameter is 
\begin_inset Formula \[
\hat{L}=\textrm{median}\left(X_{i}\right)\]

\end_inset

 where 
\begin_inset Formula $X_{i}$
\end_inset

 is a sequence of 
\begin_inset Formula $N$
\end_inset

 mutually independent Laplace RV's and the median is some number between
 the 
\begin_inset Formula $\frac{1}{2}N\textrm{th}$
\end_inset

 and the 
\begin_inset Formula $(N/2+1)\textrm{th}$
\end_inset

 order statistic (
\emph on
e.g.

\emph default
 take the average of these two) when 
\begin_inset Formula $N$
\end_inset

 is even.
 Also, 
\begin_inset Formula \[
\hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|.\]

\end_inset

 Replace 
\begin_inset Formula $\hat{L}$
\end_inset

 with 
\begin_inset Formula $L$
\end_inset

 if it is known.
 If 
\begin_inset Formula $L$
\end_inset

 is known then this estimator is distributed as 
\begin_inset Formula $\left(2N\right)^{-1}S\cdot\chi_{2N}^{2}$
\end_inset

.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(2e\right)\\
 & \approx & 1.6931471805599453094.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Left-skewed Lévy
\end_layout

\begin_layout Standard
Special case of Lévy-stable distribution with 
\begin_inset Formula $\alpha=\frac{1}{2}$
\end_inset

 and 
\begin_inset Formula $\beta=-1$
\end_inset

 the support is 
\begin_inset Formula $x<0$
\end_inset

.
 In standard form
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\left|x\right|\sqrt{2\pi\left|x\right|}}\exp\left(-\frac{1}{2\left|x\right|}\right)\\
F\left(x\right) & = & 2\Phi\left(\frac{1}{\sqrt{\left|x\right|}}\right)-1\\
G\left(q\right) & = & -\left[\Phi^{-1}\left(\frac{q+1}{2}\right)\right]^{-2}.\end{eqnarray*}

\end_inset

No moments.
\end_layout

\begin_layout Section
Lévy
\end_layout

\begin_layout Standard
A special case of Lévy-stable distributions with 
\begin_inset Formula $\alpha=\frac{1}{2}$
\end_inset

 and 
\begin_inset Formula $\beta=1$
\end_inset

.
 In standard form it is defined for 
\begin_inset Formula $x>0$
\end_inset

 as 
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{x\sqrt{2\pi x}}\exp\left(-\frac{1}{2x}\right)\\
F\left(x\right) & = & 2\left[1-\Phi\left(\frac{1}{\sqrt{x}}\right)\right]\\
G\left(q\right) & = & \left[\Phi^{-1}\left(1-\frac{q}{2}\right)\right]^{-2}.\end{eqnarray*}

\end_inset

 It has no finite moments.
\end_layout

\begin_layout Section
Logistic (Sech-squared)
\end_layout

\begin_layout Standard
A special case of the Generalized Logistic distribution with 
\begin_inset Formula $c=1.$
\end_inset

 Defined for 
\begin_inset Formula $x>0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{2}}\\
F\left(x\right) & = & \frac{1}{1+\exp\left(-x\right)}\\
G\left(q\right) & = & -\log\left(1/q-1\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \gamma+\psi_{0}\left(1\right)=0\\
\mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(1\right)=\frac{\pi^{2}}{3}\\
\gamma_{1} & = & \frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}=0\\
\gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\mu_{2}^{2}}=\frac{6}{5}\\
m_{d} & = & \log1=0\\
m_{n} & = & -\log\left(2-1\right)=0\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=1.\]

\end_inset


\end_layout

\begin_layout Section
Log Double Exponential (Log-Laplace)
\end_layout

\begin_layout Standard
Defined over 
\begin_inset Formula $x>0$
\end_inset

 with 
\begin_inset Formula $c>0$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \left\{ \begin{array}{ccc}
\frac{c}{2}x^{c-1} &  & 0<x<1\\
\frac{c}{2}x^{-c-1} &  & x\geq1\end{array}\right.\\
F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
\frac{1}{2}x^{c} &  & 0<x<1\\
1-\frac{1}{2}x^{-c} &  & x\geq1\end{array}\right.\\
G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
\left(2q\right)^{1/c} &  & 0\leq q<\frac{1}{2}\\
\left(2-2q\right)^{-1/c} &  & \frac{1}{2}\leq q\leq1\end{array}\right.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\log\left(\frac{2e}{c}\right)\]

\end_inset


\end_layout

\begin_layout Section
Log Gamma
\end_layout

\begin_layout Standard
A single shape parameter 
\begin_inset Formula $c>0$
\end_inset

 (Defined for all 
\begin_inset Formula $x$
\end_inset

) 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{\exp\left(cx-e^{x}\right)}{\Gamma\left(c\right)}\\
F\left(x;c\right) & = & \frac{\Gamma\left(c,e^{x}\right)}{\Gamma\left(c\right)}\\
G\left(q;c\right) & = & \log\left[\Gamma^{-1}\left[c,q\Gamma\left(c\right)\right]\right]\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{0}^{\infty}\left[\log y\right]^{n}y^{c-1}\exp\left(-y\right)dy.\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \mu_{1}^{\prime}\\
\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Log Normal (Cobb-Douglass)
\end_layout

\begin_layout Standard
Has one shape parameter 
\begin_inset Formula $\sigma$
\end_inset

>0.
 (Notice that the 
\begin_inset Quotes eld
\end_inset

Regress
\begin_inset Quotes erd
\end_inset

 
\begin_inset Formula $A=\log S$
\end_inset

 where 
\begin_inset Formula $S$
\end_inset

 is the scale parameter and 
\begin_inset Formula $A$
\end_inset

 is the mean of the underlying normal distribution).
 The standard form is 
\begin_inset Formula $x>0$
\end_inset


\begin_inset Formula \begin{eqnarray*}
f\left(x;\sigma\right) & = & \frac{1}{\sigma x\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{\log x}{\sigma}\right)^{2}\right]\\
F\left(x;\sigma\right) & = & \Phi\left(\frac{\log x}{\sigma}\right)\\
G\left(q;\sigma\right) & = & \exp\left\{ \sigma\Phi^{-1}\left(q\right)\right\} \end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \exp\left(\sigma^{2}/2\right)\\
\mu_{2} & = & \exp\left(\sigma^{2}\right)\left[\exp\left(\sigma^{2}\right)-1\right]\\
\gamma_{1} & = & \sqrt{p-1}\left(2+p\right)\\
\gamma_{2} & = & p^{4}+2p^{3}+3p^{2}-6\quad\quad p=e^{\sigma^{2}}\end{eqnarray*}

\end_inset

  
\end_layout

\begin_layout Standard
Notice that using JKB notation we have 
\begin_inset Formula $\theta=L,$
\end_inset

 
\begin_inset Formula $\zeta=\log S$
\end_inset

 and we have given the so-called antilognormal form of the distribution.
 This is more consistent with the location, scale parameter description
 of general probability distributions.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right].\]

\end_inset


\end_layout

\begin_layout Standard
Also, note that if 
\begin_inset Formula $X$
\end_inset

 is a log-normally distributed random-variable with 
\begin_inset Formula $L=0$
\end_inset

 and 
\begin_inset Formula $S$
\end_inset

 and shape parameter 
\begin_inset Formula $\sigma.$
\end_inset

 Then, 
\begin_inset Formula $\log X$
\end_inset

 is normally distributed with variance 
\begin_inset Formula $\sigma^{2}$
\end_inset

 and mean 
\begin_inset Formula $\log S.$
\end_inset


\end_layout

\begin_layout Section
Nakagami
\end_layout

\begin_layout Standard
Generalization of the chi distribution.
 Shape parameter is 
\begin_inset Formula $\nu>0.$
\end_inset

 Defined for 
\begin_inset Formula $x>0.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu\right) & = & \frac{2\nu^{\nu}}{\Gamma\left(\nu\right)}x^{2\nu-1}\exp\left(-\nu x^{2}\right)\\
F\left(x;\nu\right) & = & \Gamma\left(\nu,\nu x^{2}\right)\\
G\left(q;\nu\right) & = & \sqrt{\frac{1}{\nu}\Gamma^{-1}\left(v,q\right)}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\Gamma\left(\nu+\frac{1}{2}\right)}{\sqrt{\nu}\Gamma\left(\nu\right)}\\
\mu_{2} & = & \left[1-\mu^{2}\right]\\
\gamma_{1} & = & \frac{\mu\left(1-4v\mu_{2}\right)}{2\nu\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{-6\mu^{4}\nu+\left(8\nu-2\right)\mu^{2}-2\nu+1}{\nu\mu_{2}^{2}}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Noncentral beta*
\end_layout

\begin_layout Standard
Defined over 
\begin_inset Formula $x\in\left[0,1\right]$
\end_inset

 with 
\begin_inset Formula $a>0$
\end_inset

 and 
\begin_inset Formula $b>0$
\end_inset

 and 
\begin_inset Formula $c\geq0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
F\left(x;a,b,c\right)=\sum_{j=0}^{\infty}\frac{e^{-c/2}\left(\frac{c}{2}\right)^{j}}{j!}I_{B}\left(a+j,b;0\right)\]

\end_inset


\end_layout

\begin_layout Section
Noncentral chi*
\end_layout

\begin_layout Section
Noncentral chi-squared
\end_layout

\begin_layout Standard
The distribution of 
\begin_inset Formula $\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}$
\end_inset

 where 
\begin_inset Formula $Z_{i}$
\end_inset

 are independent standard normal variables and 
\begin_inset Formula $\delta_{i}$
\end_inset

 are constants.
 
\begin_inset Formula $\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.$
\end_inset

 (In communications it is called the Marcum-Q function).
 Can be thought of as a Generalized Rayleigh-Rice distribution.
 For 
\begin_inset Formula $x>0$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu,\lambda\right) & = & e^{-\left(\lambda+x\right)/2}\frac{1}{2}\left(\frac{x}{\lambda}\right)^{\left(\nu-2\right)/4}I_{\left(\nu-2\right)/2}\left(\sqrt{\lambda x}\right)\\
F\left(x;\nu,\lambda\right) & = & \sum_{j=0}^{\infty}\left\{ \frac{\left(\lambda/2\right)^{j}}{j!}e^{-\lambda/2}\right\} \textrm{Pr}\left[\chi_{\nu+2j}^{2}\leq x\right]\\
G\left(q;\nu,\lambda\right) & = & F^{-1}\left(x;\nu,\lambda\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \nu+\lambda\\
\mu_{2} & = & 2\left(\nu+2\lambda\right)\\
\gamma_{1} & = & \frac{\sqrt{8}\left(\nu+3\lambda\right)}{\left(\nu+2\lambda\right)^{3/2}}\\
\gamma_{2} & = & \frac{12\left(\nu+4\lambda\right)}{\left(\nu+2\lambda\right)^{2}}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Noncentral F
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $\lambda>0$
\end_inset

 and 
\begin_inset Formula $\nu_{1}>0$
\end_inset

 and 
\begin_inset Formula $\nu_{2}>0.$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\lambda,\nu_{1},\nu_{2}\right) & = & \exp\left[\frac{\lambda}{2}+\frac{\left(\lambda\nu_{1}x\right)}{2\left(\nu_{1}x+\nu_{2}\right)}\right]\nu_{1}^{\nu_{1}/2}\nu_{2}^{\nu_{2}/2}x^{\nu_{1}/2-1}\\
 &  & \times\left(\nu_{2}+\nu_{1}x\right)^{-\left(\nu_{1}+\nu_{2}\right)/2}\frac{\Gamma\left(\frac{\nu_{1}}{2}\right)\Gamma\left(1+\frac{\nu_{2}}{2}\right)L_{\nu_{2}/2}^{\nu_{1}/2-1}\left(-\frac{\lambda\nu_{1}x}{2\left(\nu_{1}x+\nu_{2}\right)}\right)}{B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)\Gamma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Noncentral t
\end_layout

\begin_layout Standard
The distribution of the ratio 
\begin_inset Formula \[
\frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}}\]

\end_inset

 where 
\begin_inset Formula $U$
\end_inset

 and 
\begin_inset Formula $\chi_{\nu}$
\end_inset

 are independent and distributed as a standard normal and chi with 
\begin_inset Formula $\nu$
\end_inset

 degrees of freedom.
 Note 
\begin_inset Formula $\lambda>0$
\end_inset

 and 
\begin_inset Formula $\nu>0$
\end_inset

.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\lambda,\nu\right) & = & \frac{\nu^{\nu/2}\Gamma\left(\nu+1\right)}{2^{\nu}e^{\lambda^{2}/2}\left(\nu+x^{2}\right)^{\nu/2}\Gamma\left(\nu/2\right)}\\
 &  & \times\left\{ \frac{\sqrt{2}\lambda x\,_{1}F_{1}\left(\frac{\nu}{2}+1;\frac{3}{2};\frac{\lambda^{2}x^{2}}{2\left(\nu+x^{2}\right)}\right)}{\left(\nu+x^{2}\right)\Gamma\left(\frac{\nu+1}{2}\right)}\right.\\
 &  & -\left.\frac{\,_{1}F_{1}\left(\frac{\nu+1}{2};\frac{1}{2};\frac{\lambda^{2}x^{2}}{2\left(\nu+x^{2}\right)}\right)}{\sqrt{\nu+x^{2}}\Gamma\left(\frac{\nu}{2}+1\right)}\right\} \\
 & = & \frac{\Gamma\left(\nu+1\right)}{2^{\left(\nu-1\right)/2}\sqrt{\pi\nu}\Gamma\left(\nu/2\right)}\exp\left[-\frac{\nu\lambda^{2}}{\nu+x^{2}}\right]\\
 &  & \times\left(\frac{\nu}{\nu+x^{2}}\right)^{\left(\nu-1\right)/2}Hh_{\nu}\left(-\frac{\lambda x}{\sqrt{\nu+x^{2}}}\right)\\
F\left(x;\lambda,\nu\right) & =\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Normal
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}\\
F\left(x\right) & = & \Phi\left(x\right)=\frac{1}{2}+\frac{1}{2}\textrm{erf}\left(\frac{\textrm{x}}{\sqrt{2}}\right)\\
G\left(q\right) & = & \Phi^{-1}\left(q\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Formula \begin{eqnarray*}
m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & 1\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & 0\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\
 & \approx & 1.4189385332046727418\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Maxwell
\end_layout

\begin_layout Standard
This is a special case of the Chi distribution with 
\begin_inset Formula $L=0$
\end_inset

 and 
\begin_inset Formula $S=S=\frac{1}{\sqrt{a}}$
\end_inset

 and 
\begin_inset Formula $\nu=3.$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \sqrt{\frac{2}{\pi}}x^{2}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\
F\left(x\right) & = & \Gamma\left(\frac{3}{2},\frac{x^{2}}{2}\right)\\
G\left(\alpha\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\alpha\right)}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 2\sqrt{\frac{2}{\pi}}\\
\mu_{2} & = & 3-\frac{8}{\pi}\\
\gamma_{1} & = & \sqrt{2}\frac{32-10\pi}{\left(3\pi-8\right)^{3/2}}\\
\gamma_{2} & = & \frac{-12\pi^{2}+160\pi-384}{\left(3\pi-8\right)^{2}}\\
m_{d} & = & \sqrt{2}\\
m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{3}{2},\frac{1}{2}\right)}\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=\log\left(\sqrt{\frac{2\pi}{e}}\right)+\gamma.\]

\end_inset


\end_layout

\begin_layout Section
Mielke's Beta-Kappa
\end_layout

\begin_layout Standard
A generalized F distribution.
 Two shape parameters 
\begin_inset Formula $\kappa$
\end_inset

 and 
\begin_inset Formula $\theta$
\end_inset

, and 
\begin_inset Formula $x>0$
\end_inset

.
 The 
\begin_inset Formula $\beta$
\end_inset

 in the DATAPLOT reference is a scale parameter.
\begin_inset Formula \begin{eqnarray*}
f\left(x;\kappa,\theta\right) & = & \frac{\kappa x^{\kappa-1}}{\left(1+x^{\theta}\right)^{1+\frac{\kappa}{\theta}}}\\
F\left(x;\kappa,\theta\right) & = & \frac{x^{\kappa}}{\left(1+x^{\theta}\right)^{\kappa/\theta}}\\
G\left(q;\kappa,\theta\right) & = & \left(\frac{q^{\theta/\kappa}}{1-q^{\theta/\kappa}}\right)^{1/\theta}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Pareto
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $x\geq1$
\end_inset

 and 
\begin_inset Formula $b>0$
\end_inset

.
 Standard form is 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;b\right) & = & \frac{b}{x^{b+1}}\\
F\left(x;b\right) & = & 1-\frac{1}{x^{b}}\\
G\left(q;b\right) & = & \left(1-q\right)^{-1/b}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{b}{b-1}\quad b>1\\
\mu_{2} & = & \frac{b}{\left(b-2\right)\left(b-1\right)^{2}}\quad b>2\\
\gamma_{1} & = & \frac{2\left(b+1\right)\sqrt{b-2}}{\left(b-3\right)\sqrt{b}}\quad b>3\\
\gamma_{2} & = & \frac{6\left(b^{3}+b^{2}-6b-2\right)}{b\left(b^{2}-7b+12\right)}\quad b>4\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left(X\right)=\frac{1}{c}+1-\log\left(c\right)\]

\end_inset


\end_layout

\begin_layout Section
Pareto Second Kind (Lomax)
\end_layout

\begin_layout Standard
\begin_inset Formula $c>0.$
\end_inset

 This is Pareto of the first kind with 
\begin_inset Formula $L=-1.0$
\end_inset

 so 
\begin_inset Formula $x\geq0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{c}{\left(1+x\right)^{c+1}}\\
F\left(x;c\right) & = & 1-\frac{1}{\left(1+x\right)^{c}}\\
G\left(q;c\right) & = & \left(1-q\right)^{-1/c}-1\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=\frac{1}{c}+1-\log\left(c\right).\]

\end_inset


\end_layout

\begin_layout Section
Power Log Normal
\end_layout

\begin_layout Standard
A generalization of the log-normal distribution 
\begin_inset Formula $\sigma>0$
\end_inset

 and 
\begin_inset Formula $c>0$
\end_inset

 and 
\begin_inset Formula $x>0$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;\sigma,c\right) & = & \frac{c}{x\sigma}\phi\left(\frac{\log x}{\sigma}\right)\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c-1}\\
F\left(x;\sigma,c\right) & = & 1-\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c}\\
G\left(q;\sigma,c\right) & = & \exp\left[-\sigma\Phi^{-1}\left[\left(1-q\right)^{1/c}\right]\right]\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
\mu_{n}^{\prime}=\int_{0}^{1}\exp\left[-n\sigma\Phi^{-1}\left(y^{1/c}\right)\right]dy\]

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \mu_{1}^{\prime}\\
\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}

\end_inset

 This distribution reduces to the log-normal distribution when 
\begin_inset Formula $c=1.$
\end_inset

 
\end_layout

\begin_layout Section
Power Normal
\end_layout

\begin_layout Standard
A generalization of the normal distribution, 
\begin_inset Formula $c>0$
\end_inset

 for 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & c\phi\left(x\right)\left(\Phi\left(-x\right)\right)^{c-1}\\
F\left(x;c\right) & = & 1-\left(\Phi\left(-x\right)\right)^{c}\\
G\left(q;c\right) & = & -\Phi^{-1}\left[\left(1-q\right)^{1/c}\right]\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
\mu_{n}^{\prime}=\left(-1\right)^{n}\int_{0}^{1}\left[\Phi^{-1}\left(y^{1/c}\right)\right]^{n}dy\]

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \mu_{1}^{\prime}\\
\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}

\end_inset

For 
\begin_inset Formula $c=1$
\end_inset

 this reduces to the normal distribution.
 
\end_layout

\begin_layout Section
Power-function 
\end_layout

\begin_layout Standard
A special case of the beta distribution with 
\begin_inset Formula $b=1$
\end_inset

: defined for 
\begin_inset Formula $x\in\left[0,1\right]$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
a>0\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;a\right) & = & ax^{a-1}\\
F\left(x;a\right) & = & x^{a}\\
G\left(q;a\right) & = & q^{1/a}\\
\mu & = & \frac{a}{a+1}\\
\mu_{2} & = & \frac{a\left(a+2\right)}{\left(a+1\right)^{2}}\\
\gamma_{1} & = & 2\left(1-a\right)\sqrt{\frac{a+2}{a\left(a+3\right)}}\\
\gamma_{2} & = & \frac{6\left(a^{3}-a^{2}-6a+2\right)}{a\left(a+3\right)\left(a+4\right)}\\
m_{d} & = & 1\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=1-\frac{1}{a}-\log\left(a\right)\]

\end_inset


\end_layout

\begin_layout Section
R-distribution
\end_layout

\begin_layout Standard
A general-purpose distribution with a variety of shapes controlled by 
\begin_inset Formula $c>0.$
\end_inset

 Range of standard distribution is 
\begin_inset Formula $x\in\left[-1,1\right]$
\end_inset


\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{\left(1-x^{2}\right)^{c/2-1}}{B\left(\frac{1}{2},\frac{c}{2}\right)}\\
F\left(x;c\right) & = & \frac{1}{2}+\frac{x}{B\left(\frac{1}{2},\frac{c}{2}\right)}\,_{2}F_{1}\left(\frac{1}{2},1-\frac{c}{2};\frac{3}{2};x^{2}\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
\mu_{n}^{\prime}=\frac{\left(1+\left(-1\right)^{n}\right)}{2}B\left(\frac{n+1}{2},\frac{c}{2}\right)\]

\end_inset

 The R-distribution with parameter 
\begin_inset Formula $n$
\end_inset

 is the distribution of the correlation coefficient of a random sample of
 size 
\begin_inset Formula $n$
\end_inset

 drawn from a bivariate normal distribution with 
\begin_inset Formula $\rho=0.$
\end_inset

 The mean of the standard distribution is always zero and as the sample
 size grows, the distribution's mass concentrates more closely about this
 mean.
 
\end_layout

\begin_layout Section
Rayleigh 
\end_layout

\begin_layout Standard
This is Chi distribution with 
\begin_inset Formula $L=0.0$
\end_inset

 and 
\begin_inset Formula $\nu=2$
\end_inset

 and 
\begin_inset Formula $S=S$
\end_inset

 (no location parameter is generally used), the mode of the distribution
 is 
\begin_inset Formula $S.$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(r\right) & = & re^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\
F\left(r\right) & = & 1-e^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\
G\left(q\right) & = & \sqrt{-2\log\left(1-q\right)}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \sqrt{\frac{\pi}{2}}\\
\mu_{2} & = & \frac{4-\pi}{2}\\
\gamma_{1} & = & \frac{2\left(\pi-3\right)\sqrt{\pi}}{\left(4-\pi\right)^{3/2}}\\
\gamma_{2} & = & \frac{24\pi-6\pi^{2}-16}{\left(4-\pi\right)^{2}}\\
m_{d} & = & 1\\
m_{n} & = & \sqrt{2\log\left(2\right)}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \[
h\left[X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right).\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right)\]

\end_inset


\end_layout

\begin_layout Section
Rice*
\end_layout

\begin_layout Standard
Defined for 
\begin_inset Formula $x>0$
\end_inset

 and 
\begin_inset Formula $b>0$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;b\right) & = & x\exp\left(-\frac{x^{2}+b^{2}}{2}\right)I_{0}\left(xb\right)\\
F\left(x;b\right) & = & \int_{0}^{x}\alpha\exp\left(-\frac{\alpha^{2}+b^{2}}{2}\right)I_{0}\left(\alpha b\right)d\alpha\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(1+\frac{n}{2}\right)\,_{1}F_{1}\left(-\frac{n}{2};1;-\frac{b^{2}}{2}\right)\]

\end_inset


\end_layout

\begin_layout Section
Reciprocal
\end_layout

\begin_layout Standard
Shape parameters 
\begin_inset Formula $a,b>0$
\end_inset

 
\begin_inset Formula $x\in\left[a,b\right]$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;a,b\right) & = & \frac{1}{x\log\left(b/a\right)}\\
F\left(x;a,b\right) & = & \frac{\log\left(x/a\right)}{\log\left(b/a\right)}\\
G\left(q;a,b\right) & = & a\exp\left(q\log\left(b/a\right)\right)=a\left(\frac{b}{a}\right)^{q}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
d & = & \log\left(a/b\right)\\
\mu & = & \frac{a-b}{d}\\
\mu_{2} & = & \mu\frac{a+b}{2}-\mu^{2}=\frac{\left(a-b\right)\left[a\left(d-2\right)+b\left(d+2\right)\right]}{2d^{2}}\\
\gamma_{1} & = & \frac{\sqrt{2}\left[12d\left(a-b\right)^{2}+d^{2}\left(a^{2}\left(2d-9\right)+2abd+b^{2}\left(2d+9\right)\right)\right]}{3d\sqrt{a-b}\left[a\left(d-2\right)+b\left(d+2\right)\right]^{3/2}}\\
\gamma_{2} & = & \frac{-36\left(a-b\right)^{3}+36d\left(a-b\right)^{2}\left(a+b\right)-16d^{2}\left(a^{3}-b^{3}\right)+3d^{3}\left(a^{2}+b^{2}\right)\left(a+b\right)}{3\left(a-b\right)\left[a\left(d-2\right)+b\left(d+2\right)\right]^{2}}-3\\
m_{d} & = & a\\
m_{n} & = & \sqrt{ab}\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=\frac{1}{2}\log\left(ab\right)+\log\left[\log\left(\frac{b}{a}\right)\right].\]

\end_inset


\end_layout

\begin_layout Section
Reciprocal Inverse Gaussian 
\end_layout

\begin_layout Standard
The pdf is found from the inverse gaussian (IG), 
\begin_inset Formula $f_{RIG}\left(x;\mu\right)=\frac{1}{x^{2}}f_{IG}\left(\frac{1}{x};\mu\right)$
\end_inset

 defined for 
\begin_inset Formula $x\geq0$
\end_inset

 as 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f_{IG}\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2x\mu^{2}}\right).\\
F_{IG}\left(x;\mu\right) & = & \Phi\left(\frac{1}{\sqrt{x}}\frac{x-\mu}{\mu}\right)+\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{x+\mu}{\mu}\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f_{RIG}\left(x;\mu\right) & = & \frac{1}{\sqrt{2\pi x}}\exp\left(-\frac{\left(1-\mu x\right)^{2}}{2x\mu^{2}}\right)\\
F_{RIG}\left(x;\mu\right) & = & 1-F_{IG}\left(\frac{1}{x},\mu\right)\\
 & = & 1-\Phi\left(\frac{1}{\sqrt{x}}\frac{1-\mu x}{\mu}\right)-\exp\left(\frac{2}{\mu}\right)\Phi\left(-\frac{1}{\sqrt{x}}\frac{1+\mu x}{\mu}\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Semicircular
\end_layout

\begin_layout Standard
Defined on 
\begin_inset Formula $x\in\left[-1,1\right]$
\end_inset


\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{2}{\pi}\sqrt{1-x^{2}}\\
F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\left[x\sqrt{1-x^{2}}+\arcsin x\right]\\
G\left(q\right) & = & F^{-1}\left(q\right)\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
m_{d}=m_{n}=\mu & = & 0\\
\mu_{2} & = & \frac{1}{4}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & -1\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=0.64472988584940017414.\]

\end_inset


\end_layout

\begin_layout Section
Studentized Range*
\end_layout

\begin_layout Section
Student t
\end_layout

\begin_layout Standard
Shape parameter 
\begin_inset Formula $\nu>0.$
\end_inset

 
\begin_inset Formula $I\left(a,b,x\right)$
\end_inset

 is the incomplete beta integral and 
\begin_inset Formula $I^{-1}\left(a,b,I\left(a,b,x\right)\right)=x$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu\right) & = & \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi\nu}\Gamma\left(\frac{\nu}{2}\right)\left[1+\frac{x^{2}}{\nu}\right]^{\frac{\nu+1}{2}}}\\
F\left(x;\nu\right) & = & \left\{ \begin{array}{ccc}
\frac{1}{2}I\left(\frac{\nu}{2},\frac{1}{2},\frac{\nu}{\nu+x^{2}}\right) &  & x\leq0\\
1-\frac{1}{2}I\left(\frac{\nu}{2},\frac{1}{2},\frac{\nu}{\nu+x^{2}}\right) &  & x\geq0\end{array}\right.\\
G\left(q;\nu\right) & = & \left\{ \begin{array}{ccc}
-\sqrt{\frac{\nu}{I^{-1}\left(\frac{\nu}{2},\frac{1}{2},2q\right)}-\nu} &  & q\leq\frac{1}{2}\\
\sqrt{\frac{\nu}{I^{-1}\left(\frac{\nu}{2},\frac{1}{2},2-2q\right)}-\nu} &  & q\geq\frac{1}{2}\end{array}\right.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
m_{n}=m_{d}=\mu & = & 0\\
\mu_{2} & = & \frac{\nu}{\nu-2}\quad\nu>2\\
\gamma_{1} & = & 0\quad\nu>3\\
\gamma_{2} & = & \frac{6}{\nu-4}\quad\nu>4\end{eqnarray*}

\end_inset

 As 
\begin_inset Formula $\nu\rightarrow\infty,$
\end_inset

 this distribution approaches the standard normal distribution.
 
\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\frac{1}{4}\log\left(\frac{\pi c\Gamma^{2}\left(\frac{c}{2}\right)}{\Gamma^{2}\left(\frac{c+1}{2}\right)}\right)-\frac{\left(c+1\right)}{4}\left[\Psi\left(\frac{c}{2}\right)-cZ\left(c\right)+\pi\tan\left(\frac{\pi c}{2}\right)+\gamma+2\log2\right]\]

\end_inset

where 
\begin_inset Formula \[
Z\left(c\right)=\,_{3}F_{2}\left(1,1,1+\frac{c}{2};\frac{3}{2},2;1\right)=\sum_{k=0}^{\infty}\frac{k!}{k+1}\frac{\Gamma\left(\frac{c}{2}+1+k\right)}{\Gamma\left(\frac{c}{2}+1\right)}\frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(\frac{3}{2}+k\right)}\]

\end_inset


\end_layout

\begin_layout Section
Student Z
\end_layout

\begin_layout Standard
The student Z distriubtion is defined over all space with one shape parameter
 
\begin_inset Formula $\nu>0$
\end_inset


\begin_inset Formula \begin{eqnarray*}
f\left(x;\nu\right) & = & \frac{\Gamma\left(\frac{\nu}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{\nu-1}{2}\right)}\left(1+x^{2}\right)^{-\nu/2}\\
F\left(x;\nu\right) & = & \left\{ \begin{array}{ccc}
Q\left(x;\nu\right) &  & x\leq0\\
1-Q\left(x;\nu\right) &  & x\geq0\end{array}\right.\\
Q\left(x;\nu\right) & = & \frac{\left|x\right|^{1-n}\Gamma\left(\frac{n}{2}\right)\,_{2}F_{1}\left(\frac{n-1}{2},\frac{n}{2};\frac{n+1}{2};-\frac{1}{x^{2}}\right)}{2\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}\end{eqnarray*}

\end_inset

Interesting moments are 
\begin_inset Formula \begin{eqnarray*}
\mu & = & 0\\
\sigma^{2} & = & \frac{1}{\nu-3}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{6}{\nu-5}.\end{eqnarray*}

\end_inset

 The moment generating function is 
\begin_inset Formula \[
\theta\left(t\right)=2\sqrt{\left|\frac{t}{2}\right|^{\nu-1}}\frac{K_{\left(n-1\right)/2}\left(\left|t\right|\right)}{\Gamma\left(\frac{\nu-1}{2}\right)}.\]

\end_inset


\end_layout

\begin_layout Section
Symmetric Power*
\end_layout

\begin_layout Section
Triangular
\end_layout

\begin_layout Standard
One shape parameter 
\begin_inset Formula $c\in[0,1]$
\end_inset

 giving the distance to the peak as a percentage of the total extent of
 the non-zero portion.
 The location parameter is the start of the non-zero portion, and the scale-para
meter is the width of the non-zero portion.
 In standard form we have 
\begin_inset Formula $x\in\left[0,1\right].$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \left\{ \begin{array}{ccc}
2\frac{x}{c} &  & x<c\\
2\frac{1-x}{1-c} &  & x\geq c\end{array}\right.\\
F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
\frac{x^{2}}{c} &  & x<c\\
\frac{x^{2}-2x+c}{c-1} &  & x\geq c\end{array}\right.\\
G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
\sqrt{cq} &  & q<c\\
1-\sqrt{\left(1-c\right)\left(1-q\right)} &  & q\geq c\end{array}\right.\end{eqnarray*}

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{c}{3}+\frac{1}{3}\\
\mu_{2} & = & \frac{1-c+c^{2}}{18}\\
\gamma_{1} & = & \frac{\sqrt{2}\left(2c-1\right)\left(c+1\right)\left(c-2\right)}{5\left(1-c+c^{2}\right)^{3/2}}\\
\gamma_{2} & = & -\frac{3}{5}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
h\left(X\right) & = & \log\left(\frac{1}{2}\sqrt{e}\right)\\
 & \approx & -0.19314718055994530942.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Truncated Exponential
\end_layout

\begin_layout Standard
This is an exponential distribution defined only over a certain region 
\begin_inset Formula $0<x<B$
\end_inset

.
 In standard form this is 
\begin_inset Formula \begin{eqnarray*}
f\left(x;B\right) & = & \frac{e^{-x}}{1-e^{-B}}\\
F\left(x;B\right) & = & \frac{1-e^{-x}}{1-e^{-B}}\\
G\left(q;B\right) & = & -\log\left(1-q+qe^{-B}\right)\end{eqnarray*}

\end_inset


\begin_inset Formula \[
\mu_{n}^{\prime}=\Gamma\left(1+n\right)-\Gamma\left(1+n,B\right)\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \[
h\left[X\right]=\log\left(e^{B}-1\right)+\frac{1+e^{B}\left(B-1\right)}{1-e^{B}}.\]

\end_inset


\end_layout

\begin_layout Section
Truncated Normal
\end_layout

\begin_layout Standard
A normal distribution restricted to lie within a certain range given by
 two parameters 
\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $B$
\end_inset

.
 Notice that this 
\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $B$
\end_inset

 correspond to the bounds on 
\begin_inset Formula $x$
\end_inset

 in standard form.
 For 
\begin_inset Formula $x\in\left[A,B\right]$
\end_inset

 we get 
\begin_inset Formula \begin{eqnarray*}
f\left(x;A,B\right) & = & \frac{\phi\left(x\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
F\left(x;A,B\right) & = & \frac{\Phi\left(x\right)-\Phi\left(A\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
G\left(q;A,B\right) & = & \Phi^{-1}\left[q\Phi\left(B\right)+\Phi\left(A\right)\left(1-q\right)\right]\end{eqnarray*}

\end_inset

 where 
\begin_inset Formula \begin{eqnarray*}
\phi\left(x\right) & = & \frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\\
\Phi\left(x\right) & = & \int_{-\infty}^{x}\phi\left(u\right)du.\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\
\mu_{2} & = & 1+\frac{A\phi\left(A\right)-B\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}-\left(\frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\right)^{2}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Tukey-Lambda
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x;\lambda\right) & = & F^{\prime}\left(x;\lambda\right)=\frac{1}{G^{\prime}\left(F\left(x;\lambda\right);\lambda\right)}=\frac{1}{F^{\lambda-1}\left(x;\lambda\right)+\left[1-F\left(x;\lambda\right)\right]^{\lambda-1}}\\
F\left(x;\lambda\right) & = & G^{-1}\left(x;\lambda\right)\\
G\left(p;\lambda\right) & = & \frac{p^{\lambda}-\left(1-p\right)^{\lambda}}{\lambda}\end{eqnarray*}

\end_inset

 
\begin_inset Formula \begin{eqnarray*}
\mu & = & 0\\
\mu_{2} & = & \int_{0}^{1}G^{2}\left(p;\lambda\right)dp\\
 & = & \frac{2\Gamma\left(\lambda+\frac{3}{2}\right)-\lambda4^{-\lambda}\sqrt{\pi}\Gamma\left(\lambda\right)\left(1-2\lambda\right)}{\lambda^{2}\left(1+2\lambda\right)\Gamma\left(\lambda+\frac{3}{2}\right)}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{\mu_{4}}{\mu_{2}^{2}}-3\\
\mu_{4} & = & \frac{3\Gamma\left(\lambda\right)\Gamma\left(\lambda+\frac{1}{2}\right)2^{-2\lambda}}{\lambda^{3}\Gamma\left(2\lambda+\frac{3}{2}\right)}+\frac{2}{\lambda^{4}\left(1+4\lambda\right)}\\
 &  & -\frac{2\sqrt{3}\Gamma\left(\lambda\right)2^{-6\lambda}3^{3\lambda}\Gamma\left(\lambda+\frac{1}{3}\right)\Gamma\left(\lambda+\frac{2}{3}\right)}{\lambda^{3}\Gamma\left(2\lambda+\frac{3}{2}\right)\Gamma\left(\lambda+\frac{1}{2}\right)}.\end{eqnarray*}

\end_inset

 Notice that the 
\begin_inset Formula $\lim_{\lambda\rightarrow0}G\left(p;\lambda\right)=\log\left(p/\left(1-p\right)\right)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
h\left[X\right] & = & \int_{0}^{1}\log\left[G^{\prime}\left(p\right)\right]dp\\
 & = & \int_{0}^{1}\log\left[p^{\lambda-1}+\left(1-p\right)^{\lambda-1}\right]dp.\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Uniform
\end_layout

\begin_layout Standard
Standard form 
\begin_inset Formula $x\in\left(0,1\right).$
\end_inset

 In general form, the lower limit is 
\begin_inset Formula $L,$
\end_inset

 the upper limit is 
\begin_inset Formula $S+L.$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & 1\\
F\left(x\right) & = & x\\
G\left(q\right) & = & q\end{eqnarray*}

\end_inset


\begin_inset Formula \begin{eqnarray*}
\mu & = & \frac{1}{2}\\
\mu_{2} & = & \frac{1}{12}\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & -\frac{6}{5}\end{eqnarray*}

\end_inset


\begin_inset Formula \[
h\left[X\right]=0\]

\end_inset


\end_layout

\begin_layout Section
Von Mises
\end_layout

\begin_layout Standard
Defined for 
\begin_inset Formula $x\in\left[-\pi,\pi\right]$
\end_inset

 with shape parameter 
\begin_inset Formula $b>0$
\end_inset

.
 Note, the PDF and CDF functions are periodic and are always defined over
 
\begin_inset Formula $x\in\left[-\pi,\pi\right]$
\end_inset

 regardless of the location parameter.
 Thus, if an input beyond this range is given, it is converted to the equivalent
 angle in this range.
 For values of 
\begin_inset Formula $b<100$
\end_inset

 the PDF and CDF formulas below are used.
 Otherwise, a normal approximation with variance 
\begin_inset Formula $1/b$
\end_inset

 is used.
 
\begin_inset Formula \begin{eqnarray*}
f\left(x;b\right) & = & \frac{e^{b\cos x}}{2\pi I_{0}\left(b\right)}\\
F\left(x;b\right) & = & \frac{1}{2}+\frac{x}{2\pi}+\sum_{k=1}^{\infty}\frac{I_{k}\left(b\right)\sin\left(kx\right)}{I_{0}\left(b\right)\pi k}\\
G\left(q;b\right) & = & F^{-1}\left(x;b\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 0\\
\mu_{2} & = & \int_{-\pi}^{\pi}x^{2}f\left(x;b\right)dx\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{\int_{-\pi}^{\pi}x^{4}f\left(x;b\right)dx}{\mu_{2}^{2}}-3\end{eqnarray*}

\end_inset

 This can be used for defining circular variance.
 
\end_layout

\begin_layout Section
Wald 
\end_layout

\begin_layout Standard
Special case of the Inverse Normal with shape parameter set to 
\begin_inset Formula $1.0$
\end_inset

.
 Defined for 
\begin_inset Formula $x>0$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
f\left(x\right) & = & \frac{1}{\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2x}\right).\\
F\left(x\right) & = & \Phi\left(\frac{x-1}{\sqrt{x}}\right)+\exp\left(2\right)\Phi\left(-\frac{x+1}{\sqrt{x}}\right)\\
G\left(q;\mu\right) & = & F^{-1}\left(q;\mu\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mu & = & 1\\
\mu_{2} & = & 1\\
\gamma_{1} & = & 3\\
\gamma_{2} & = & 15\\
m_{d} & = & \frac{1}{2}\left(\sqrt{13}-3\right)\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Wishart*
\end_layout

\begin_layout Section
Wrapped Cauchy
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $x\in\left[0,2\pi\right]$
\end_inset

 
\begin_inset Formula $c\in\left(0,1\right)$
\end_inset

 
\begin_inset Formula \begin{eqnarray*}
f\left(x;c\right) & = & \frac{1-c^{2}}{2\pi\left(1+c^{2}-2c\cos x\right)}\\
g_{c}\left(x\right) & = & \frac{1}{\pi}\arctan\left[\frac{1+c}{1-c}\tan\left(\frac{x}{2}\right)\right]\\
r_{c}\left(q\right) & = & 2\arctan\left[\frac{1-c}{1+c}\tan\left(\pi q\right)\right]\\
F\left(x;c\right) & = & \left\{ \begin{array}{ccc}
g_{c}\left(x\right) &  & 0\leq x<\pi\\
1-g_{c}\left(2\pi-x\right) &  & \pi\leq x\leq2\pi\end{array}\right.\\
G\left(q;c\right) & = & \left\{ \begin{array}{ccc}
r_{c}\left(q\right) &  & 0\leq q<\frac{1}{2}\\
2\pi-r_{c}\left(1-q\right) &  & \frac{1}{2}\leq q\leq1\end{array}\right.\end{eqnarray*}

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\begin_inset Formula \[
\]

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\begin_inset Formula \[
h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).\]

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