Continuous Statistical Distributions¶

Overview¶

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 $L=0.0$ and $S=1.0.$ The nonstandard forms can be obtained for the various functions using (note $U$ is a standard uniform random variate).

Function Name	Standard Function	Transformation
Cumulative Distribution Function (CDF)	$F\left(x\right)$	$F\left(x;L,S\right)=F\left(\frac{\left(x-L\right)}{S}\right)$
Probability Density Function (PDF)	$f\left(x\right)=F^{\prime}\left(x\right)$	$f\left(x;L,S\right)=\frac{1}{S}f\left(\frac{\left(x-L\right)}{S}\right)$
Percent Point Function (PPF)	$G\left(q\right)=F^{-1}\left(q\right)$	$G\left(q;L,S\right)=L+SG\left(q\right)$
Probability Sparsity Function (PSF)	$g\left(q\right)=G^{\prime}\left(q\right)$	$g\left(q;L,S\right)=Sg\left(q\right)$
Hazard Function (HF)	$h_{a}\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$	$h_{a}\left(x;L,S\right)=\frac{1}{S}h_{a}\left(\frac{\left(x-L\right)}{S}\right)$
Cumulative Hazard Functon (CHF)	$H_{a}\left(x\right)=$ $\log\frac{1}{1-F\left(x\right)}$	$H_{a}\left(x;L,S\right)=H_{a}\left(\frac{\left(x-L\right)}{S}\right)$
Survival Function (SF)	$S\left(x\right)=1-F\left(x\right)$	$S\left(x;L,S\right)=S\left(\frac{\left(x-L\right)}{S}\right)$
Inverse Survival Function (ISF)	$Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$	$Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)$
Moment Generating Function (MGF)	$M_{Y}\left(t\right)=E\left[e^{Yt}\right]$	$M_{X}\left(t\right)=e^{Lt}M_{Y}\left(St\right)$
Random Variates	$Y=G\left(U\right)$	$X=L+SY$
(Differential) Entropy	$h\left[Y\right]=-\int f\left(y\right)\log f\left(y\right)dy$	$h\left[X\right]=h\left[Y\right]+\log S$
(Non-central) Moments	$\mu_{n}^{\prime}=E\left[Y^{n}\right]$	$E\left[X^{n}\right]=L^{n}\sum_{k=0}^{N}\left(\begin{array}{c} n\\ k\end{array}\right)\left(\frac{S}{L}\right)^{k}\mu_{k}^{\prime}$
Central Moments	$\mu_{n}=E\left[\left(Y-\mu\right)^{n}\right]$	$E\left[\left(X-\mu_{X}\right)^{n}\right]=S^{n}\mu_{n}$
mean (mode, median), var	$\mu,\,\mu_{2}$	$L+S\mu,\, S^{2}\mu_{2}$
skewness, kurtosis	$\gamma_{1}=\frac{\mu_{3}}{\left(\mu_{2}\right)^{3/2}},\,$ $\gamma_{2}=\frac{\mu_{4}}{\left(\mu_{2}\right)^{2}}-3$	$\gamma_{1},\,\gamma_{2}$

Moments¶

Non-central moments are defined using the PDF

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Note, that these can always be computed using the PPF. Substitute $x=G\left(q\right)$ in the above equation and get

System Message: WARNING/2 ( \mu_{n}^{\prime}=\int_{0}^{1}G^{n}\left(q\right)dq)

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which may be easier to compute numerically. Note that $q=F\left(x\right)$ so that $dq=f\left(x\right)dx.$ Central moments are computed similarly $\mu=\mu_{1}^{\prime}$

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

In particular

$\begin{eqnarray*} \mu_{3} & = & \mu_{3}^{\prime}-3\mu\mu_{2}^{\prime}+2\mu^{3}\\ & = & \mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}\\ \mu_{4} & = & \mu_{4}^{\prime}-4\mu\mu_{3}^{\prime}+6\mu^{2}\mu_{2}^{\prime}-3\mu^{4}\\ & = & \mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}\end{eqnarray*}$

Skewness is defined as

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while (Fisher) kurtosis is

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so that a normal distribution has a kurtosis of zero.

Median and mode¶

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

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In addition, the mode, $m_{d}$ , is defined as the value for which the probability density function reaches it’s peak

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Fitting data¶

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.

If $f\left(x;\boldsymbol{\theta}\right)$ is the PDF of a random-variable where $\boldsymbol{\theta}$ is a vector of parameters ( e.g. $L$ and $S$ ), then for a collection of $N$ independent samples from this distribution, the joint distribution the random vector $\mathbf{x}$ is

System Message: WARNING/2 ( f\left(\mathbf{x};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f\left(x_{i};\boldsymbol{\theta}\right).)

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The maximum likelihood estimate of the parameters $\boldsymbol{\theta}$ are the parameters which maximize this function with $\mathbf{x}$ fixed and given by the data:

$\begin{eqnarray*} \boldsymbol{\theta}_{es} & = & \arg\max_{\boldsymbol{\theta}}f\left(\mathbf{x};\boldsymbol{\theta}\right)\\ & = & \arg\min_{\boldsymbol{\theta}}l_{\mathbf{x}}\left(\boldsymbol{\theta}\right).\end{eqnarray*}$

Where

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

Note that if $\boldsymbol{\theta}$ includes only shape parameters, the location and scale-parameters can be fit by replacing $x_{i}$ with $\left(x_{i}-L\right)/S$ in the log-likelihood function adding $N\log S$ and minimizing, thus

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

If desired, sample estimates for $L$ and $S$ (not necessarily maximum likelihood estimates) can be obtained from samples estimates of the mean and variance using

$\begin{eqnarray*} \hat{S} & = & \sqrt{\frac{\hat{\mu}_{2}}{\mu_{2}}}\\ \hat{L} & = & \hat{\mu}-\hat{S}\mu\end{eqnarray*}$

where $\mu$ and $\mu_{2}$ are assumed known as the mean and variance of the untransformed distribution (when $L=0$ and $S=1$ ) and

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

Standard notation for mean¶

We will use

System Message: WARNING/2 ( \overline{y\left(\mathbf{x}\right)}=\frac{1}{N}\sum_{i=1}^{N}y\left(x_{i}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \overline{y\left(\mathbf{x}\right)} =\frac{1}{N}\sum_{i=1}^{N}y\left(x_... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 556 bytes). Transcript written on math.log.

where $N$ should be clear from context as the number of samples $x_{i}$

References¶

Documentation for ranlib, rv2, cdflib
Eric Weisstein~s world of mathematics http://mathworld.wolfram.com/, http://mathworld.wolfram.com/topics/StatisticalDistributions.html
Documentation to Regress+ by Michael McLaughlin item Engineering and Statistics Handbook (NIST), http://www.itl.nist.gov/div898/handbook/index.htm
Documentation for DATAPLOT from NIST, http://www.itl.nist.gov/div898/software/dataplot/distribu.htm
Norman Johnson, Samuel Kotz, and N. Balakrishnan Continuous Univariate Distributions, second edition, Volumes I and II, Wiley & Sons, 1994.

Alpha¶

One shape parameters $\alpha>0$ (paramter $\beta$ in DATAPLOT is a scale-parameter). Standard form is $x>0:$

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

System Message: WARNING/2 ( 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)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\frac{1}{\Phi\left(a\right)\sqrt{2\pi}}\int_{0}^{\inf... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 920 bytes). Transcript written on math.log.

No moments?

System Message: WARNING/2 ( 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}})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 l_ {\mathbf{x}}\left(\alpha\right)=N\log\left[\Phi\left(\alpha\right)\... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 916 bytes). Transcript written on math.log.

Anglit¶

Defined over $x\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$

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

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

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

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

System Message: WARNING/2 ( l_{\mathbf{x}}\left(\cdot\right)=-N\overline{\log\left[\cos\left(2\mathbf{x}\right)\right]})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 l_ {\mathbf{x}}\left(\cdot\right)=-N\overline{\log\left[\cos\left(2\ma... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 468 bytes). Transcript written on math.log.

Arcsine¶

Defined over $x\in\left(0,1\right)$ . To get the JKB definition put $x=\frac{u+1}{2}.$ i.e. $L=-1$ and $S=2.$

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

System Message: WARNING/2 ( M\left(t\right)=E^{t/2}I_{0}\left(\frac{t}{2}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=E^{t/2}I_{0}\left(\frac{t}{2}\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 472 bytes). Transcript written on math.log.

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

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

System Message: WARNING/2 ( h\left[X\right]\approx-0.24156447527049044468)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]\approx-0.24156447527049044468 (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 320 bytes). Transcript written on math.log.

System Message: WARNING/2 ( 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)})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 l_ {\mathbf{x}}\left(\cdot\right)=N\log\pi+\frac{N}{2}\overline{\log\m... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 688 bytes). Transcript written on math.log.

Beta¶

Two shape parameters

$a,b>0$

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

$f\left(x;a,1\right)$ is also called the Power-function distribution.

System Message: WARNING/2 ( 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)})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 l_ {\mathbf{x}}\left(a,b\right)=-N\log\Gamma\left(a+b\right)+N\log\Gam... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 700 bytes). Transcript written on math.log.

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

Beta Prime¶

Defined over $0<x<\infty.$ $\alpha,\beta>0.$ (Note the CDF evaluation uses Eq. 3.194.1 on pg. 313 of Gradshteyn & Ryzhik (sixth edition).

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

System Message: WARNING/2 ( \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.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\left\{ \begin{array}{ccc} \frac{\Gamma\left(n+\alph... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 816 bytes). Transcript written on math.log.

Therefore,

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

Bradford¶

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

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

where $\textrm{Ei}\left(\textrm{z}\right)$ is the exponential integral function. Also

System Message: WARNING/2 ( h\left[X\right]=\frac{1}{2}\log\left(1+c\right)-\log\left(\frac{c}{\log\left(1+c\right)}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\frac{1}{2}\log\left(1+c\right)-\log\left(\frac{c}{\l... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 604 bytes). Transcript written on math.log.

Burr¶

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

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

Cauchy¶

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

No finite moments. This is the t distribution with one degree of freedom.

$\begin{eqnarray*} h\left[X\right] & = & \log\left(4\pi\right)\\ & \approx & 2.5310242469692907930.\end{eqnarray*}$

Chi¶

Generated by taking the (positive) square-root of chi-squared variates.

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

System Message: WARNING/2 ( 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))

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

Chi-squared¶

This is the gamma distribution with $L=0.0$ and $S=2.0$ and $\alpha=\nu/2$ where $\nu$ is called the degrees of freedom. If $Z_{1}\ldots Z_{\nu}$ are all standard normal distributions, then $W=\sum_{k}Z_{k}^{2}$ has (standard) chi-square distribution with $\nu$ degrees of freedom.

The standard form (most often used in standard form only) is $x>0$

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

System Message: WARNING/2 ( M\left(t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1}{2}-t\right)^{\nu/2}})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 640 bytes). Transcript written on math.log.

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

Cosine¶

Approximation to the normal distribution.

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

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

Double Gamma¶

The double gamma is the signed version of the Gamma distribution. For $\alpha>0:$

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

System Message: WARNING/2 ( M\left(t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\right)^{a}})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\r... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 556 bytes). Transcript written on math.log.

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

Doubly Non-central F*¶

Doubly Non-central t*¶

Double Weibull¶

This is a signed form of the Weibull distribution.

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\mu_{n}=\begin{cases} \Gamma\left(1+\frac{n}{c}\right) & n\textrm{ even}\\ 0 & n\textrm{ odd}\end{cases})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\mu_{n}=\begin{cases} \Gamma\left(1+\frac{n}{c}\righ... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 560 bytes). Transcript written on math.log.

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

Erlang¶

This is just the Gamma distribution with shape parameter $\alpha=n$ an integer.

Exponential¶

This is a special case of the Gamma (and Erlang) distributions with shape parameter $\left(\alpha=1\right)$ and the same location and scale parameters. The standard form is therefore ( $x\geq0$ )

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=n!)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=n! (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 336 bytes). Transcript written on math.log.

System Message: WARNING/2 ( M\left(t\right)=\frac{1}{1-t})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\frac{1}{1-t} (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 412 bytes). Transcript written on math.log.

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

System Message: WARNING/2 ( h\left[X\right]=1.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=1. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 252 bytes). Transcript written on math.log.

Exponentiated Weibull¶

Two positive shape parameters $a$ and $c$ and $x\in\left(0,\infty\right)$

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

Exponential Power¶

One positive shape parameter $b$ . Defined for $x\geq0.$

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

Fatigue Life (Birnbaum-Sanders)¶

This distribution’s pdf is the average of the inverse-Gaussian $\left(\mu=1\right)$ and reciprocal inverse-Gaussian pdf $\left(\mu=1\right)$ . We follow the notation of JKB here with $\beta=S.$ for $x>0$

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

System Message: WARNING/2 ( 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))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=c\sqrt{2\pi}\exp\left[\frac{1}{c^{2}}\left(1-\sqrt{1-... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 856 bytes). Transcript written on math.log.

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

Fisk (Log Logistic)¶

Special case of the Burr distribution with $d=1$

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

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

System Message: WARNING/2 ( h\left[X\right]=2-\log c.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=2-\log c. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 312 bytes). Transcript written on math.log.

Folded Cauchy¶

This formula can be expressed in terms of the standard formulas for the Cauchy distribution (call the cdf $C\left(x\right)$ and the pdf $d\left(x\right)$ ). if $Y$ is cauchy then $\left|Y\right|$ is folded cauchy. Note that $x\geq0.$

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

No moments

Folded Normal¶

If $Z$ is Normal with mean $L$ and $\sigma=S$ , then $\left|Z\right|$ is a folded normal with shape parameter $c=\left|L\right|/S$ , location parameter $0$ and scale parameter $S$ . This is a special case of the non-central chi distribution with one- degree of freedom and non-centrality parameter $c^{2}.$ Note that $c\geq0$ . The standard form of the folded normal is

$\begin{eqnarray*} f\left(x;c\right) & = & \sqrt{\frac{2}{\pi}}\cosh\left(cx\right)\exp\left(-\frac{x^{2}+c^{2}}{2}\right)\\ F\left(x;c\right) & = & \Phi\left(x-c\right)-\Phi\left(-x-c\right)=\Phi\left(x-c\right)+\Phi\left(x+c\right)-1\\ G\left(\alpha;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}$

System Message: WARNING/2 ( M\left(t\right)=\exp\left[\frac{t}{2}\left(t-2c\right)\right]\left(1+e^{2ct}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\exp\left[\frac{t}{2}\left(t-2c\right)\right]\left(1+... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 560 bytes). Transcript written on math.log.

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

Fratio (or F)¶

Defined for $x>0$ . The distribution of $\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)$ if $X_{1}$ is chi-squared with $v_{1}$ degrees of freedom and $X_{2}$ is chi-squared with $v_{2}$ degrees of freedom.

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

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

Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)¶

A type of extreme-value distribution with a lower bound. Defined for $x>0$ and $c>0$

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 460 bytes). Transcript written on math.log.

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

System Message: WARNING/2 ( h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1 (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 436 bytes). Transcript written on math.log.

where $\gamma$ is Euler’s constant and equal to

System Message: WARNING/2 ( \gamma\approx0.57721566490153286061.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \gamma \approx0.57721566490153286061. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 300 bytes). Transcript written on math.log.

Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)¶

Defined for $x<0$ and $c>0$ .

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

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right).)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right). (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 544 bytes). Transcript written on math.log.

System Message: WARNING/2 ( h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1 (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 436 bytes). Transcript written on math.log.

where $\gamma$ is Euler’s constant and equal to

System Message: WARNING/2 ( \gamma\approx0.57721566490153286061.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \gamma \approx0.57721566490153286061. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 300 bytes). Transcript written on math.log.

Gamma¶

The standard form for the gamma distribution is $\left(\alpha>0\right)$ valid for $x\geq0$ .

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

System Message: WARNING/2 ( M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\frac{1}{\left(1-t\right)^{\alpha}} (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 476 bytes). Transcript written on math.log.

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

System Message: WARNING/2 ( h\left[X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(a\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 384 bytes). Transcript written on math.log.

where

System Message: WARNING/2 ( \Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \Psi \left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 464 bytes). Transcript written on math.log.

Generalized Logistic¶

Has been used in the analysis of extreme values. Has one shape parameter $c>0.$ And $x>0$

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

System Message: WARNING/2 ( M\left(t\right)=\frac{c}{1-t}\,_{2}F_{1}\left(1+c,\,1-t\,;\,2-t\,;-1\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\frac{c}{1-t}\,_{2}F_{1}\left(1+c,\,1-t\,;\,2-t\,;-1\... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 544 bytes). Transcript written on math.log.

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

Note that the polygamma function is

$\begin{eqnarray*} \psi_{n}\left(z\right) & = & \frac{d^{n+1}}{dz^{n+1}}\log\Gamma\left(z\right)\\ & = & \left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\ & = & \left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\end{eqnarray*}$

where $\zeta\left(k,x\right)$ is a generalization of the Riemann zeta function called the Hurwitz zeta function Note that $\zeta\left(n\right)\equiv\zeta\left(n,1\right)$

Generalized Pareto¶

Shape parameter $c\neq0$ and defined for $x\geq0$ for all $c$ and $x<\frac{1}{\left|c\right|}$ if $c$ is negative.

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

System Message: WARNING/2 ( 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.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\left\{ \begin{array}{cc} \left(-\frac{t}{c}\right)^{... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 1492 bytes). Transcript written on math.log.

System Message: WARNING/2 ( \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)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\frac{\left(-1\right)^{n}}{c^{n}}\sum_{k=0}^{n}\left... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 728 bytes). Transcript written on math.log.

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

Thus,

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

System Message: WARNING/2 ( h\left[X\right]=1+c\quad c>0)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=1+c\quad c>0 (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 268 bytes). Transcript written on math.log.

Generalized Exponential¶

Three positive shape parameters for $x\geq0.$ Note that $a,b,$ and $c$ are all $>0.$

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

Generalized Extreme Value¶

Extreme value distributions with shape parameter $c$ .

For $c>0$ defined on $-\infty<x\leq1/c.$

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

System Message: WARNING/2 ( \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)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\frac{1}{c^{n}}\sum_{k=0}^{n}\left(\begin{array}{c} ... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 732 bytes). Transcript written on math.log.

So,

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

For $c<0$ defined on $\frac{1}{c}\leq x<\infty.$ For $c=0$ defined over all space

$\begin{eqnarray*} f\left(x;0\right) & = & \exp\left[-e^{-x}\right]e^{-x}\\ F\left(x;0\right) & = & \exp\left[-e^{-x}\right]\\ G\left(q;0\right) & = & -\log\left(-\log q\right)\end{eqnarray*}$

This is just the (left-skewed) Gumbel distribution for c=0.

$\begin{eqnarray*} \mu & = & \gamma=-\psi_{0}\left(1\right)\\ \mu_{2} & = & \frac{\pi^{2}}{6}\\ \gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\ \gamma_{2} & = & \frac{12}{5}\end{eqnarray*}$

Generalized Gamma¶

A general probability form that reduces to many common distributions: $x>0$ $a>0$ and $c\neq0.$

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 548 bytes). Transcript written on math.log.

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

Special cases are Weibull $\left(a=1\right)$ , half-normal $\left(a=1/2,c=2\right)$ and ordinary gamma distributions $c=1.$ If $c=-1$ then it is the inverted gamma distribution.

System Message: WARNING/2 ( 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|.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 552 bytes). Transcript written on math.log.

Generalized Half-Logistic¶

For $x\in\left[0,1/c\right]$ and $c>0$ we have

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

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

Gilbrat¶

Special case of the log-normal with $\sigma=1$ and $S=1.0$ (typically also $L=0.0$ )

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

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

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

Gompertz (Truncated Gumbel)¶

For $x\geq0$ and $c>0$ . In JKB the two shape parameters $b,a$ are reduced to the single shape-parameter $c=b/a$ . As $a$ is just a scale parameter when $a\neq0$ . If $a=0,$ the distribution reduces to the exponential distribution scaled by $1/b.$ Thus, the standard form is given as

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

System Message: WARNING/2 ( h\left[X\right]=1-\log\left(c\right)-e^{c}\textrm{Ei}\left(1,c\right),)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=1-\log\left(c\right)-e^{c}\textrm{Ei}\left(1,c\right), (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 428 bytes). Transcript written on math.log.

where

System Message: WARNING/2 ( \textrm{Ei}\left(n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\right)dt)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \textrm{Ei}\left (n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\righ... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 552 bytes). Transcript written on math.log.

Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)¶

One of a clase of extreme value distributions (right-skewed).

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

System Message: WARNING/2 ( M\left(t\right)=\Gamma\left(1-t\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\Gamma\left(1-t\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 316 bytes). Transcript written on math.log.

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

System Message: WARNING/2 ( h\left[X\right]\approx1.0608407169541684911)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]\approx1.0608407169541684911 (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 316 bytes). Transcript written on math.log.

Gumbel Left-skewed (for minimum order statistic)¶

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

System Message: WARNING/2 ( M\left(t\right)=\Gamma\left(1+t\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\Gamma\left(1+t\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 272 bytes). Transcript written on math.log.

Note, that $\mu$ is negative the mean for the right-skewed distribution. Similar for median and mode. All other moments are the same.

System Message: WARNING/2 ( h\left[X\right]\approx1.0608407169541684911.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]\approx1.0608407169541684911. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 320 bytes). Transcript written on math.log.

HalfCauchy¶

If $Z$ is Hyperbolic Secant distributed then $e^{Z}$ is Half-Cauchy distributed. Also, if $W$ is (standard) Cauchy distributed, then $\left|W\right|$ is Half-Cauchy distributed. Special case of the Folded Cauchy distribution with $c=0.$ The standard form is

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

System Message: WARNING/2 ( 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])

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\cos t+\frac{2}{\pi}\left[\textrm{Si}\left(t\right)\c... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 544 bytes). Transcript written on math.log.

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

No moments, as the integrals diverge.

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

HalfNormal¶

This is a special case of the chi distribution with $L=a$ and $S=b$ and $\nu=1.$ This is also a special case of the folded normal with shape parameter $c=0$ and $S=S.$ If $Z$ is (standard) normally distributed then, $\left|Z\right|$ is half-normal. The standard form is

$\begin{eqnarray*} f\left(x\right) & = & \sqrt{\frac{2}{\pi}}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\ F\left(x\right) & = & 2\Phi\left(x\right)-1\\ G\left(q\right) & = & \Phi^{-1}\left(\frac{1+q}{2}\right)\end{eqnarray*}$

System Message: WARNING/2 ( M\left(t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 504 bytes). Transcript written on math.log.

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

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

Half-Logistic¶

In the limit as $c\rightarrow\infty$ for the generalized half-logistic we have the half-logistic defined over $x\geq0.$ Also, the distribution of $\left|X\right|$ where $X$ has logistic distribtution.

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

System Message: WARNING/2 ( 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))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=1-t\psi_{0}\left(\frac{1}{2}-\frac{t}{2}\right)+t\psi... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 648 bytes). Transcript written on math.log.

System Message: WARNING/2 ( \mu_{n}^{\prime}=2\left(1-2^{1-n}\right)n!\zeta\left(n\right)\quad n\neq1)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=2\left(1-2^{1-n}\right)n!\zeta\left(n\right)\quad n\... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 504 bytes). Transcript written on math.log.

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

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

Hyperbolic Secant¶

Related to the logistic distribution and used in lifetime analysis. Standard form is (defined over all $x$ )

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

System Message: WARNING/2 ( M\left(t\right)=\sec\left(\frac{\pi}{2}t\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 M\left (t\right)=\sec\left(\frac{\pi}{2}t\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 448 bytes). Transcript written on math.log.

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

where $C_{m}$ is an integer given by

$\begin{eqnarray*} C_{m} & = & \frac{\left(2m\right)!\left[\zeta\left(2m+1,\frac{1}{4}\right)-\zeta\left(2m+1,\frac{3}{4}\right)\right]}{\pi^{2m+1}2^{2m}}\\ & = & 4\left(-1\right)^{m-1}\frac{16^{m}}{2m+1}B_{2m+1}\left(\frac{1}{4}\right)\end{eqnarray*}$

where $B_{2m+1}\left(\frac{1}{4}\right)$ is the Bernoulli polynomial of order $2m+1$ evaluated at $1/4.$ Thus

System Message: WARNING/2 ( \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.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\left\{ \begin{array}{cc} 0 & n\textrm{ odd}\\ 4\lef... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 792 bytes). Transcript written on math.log.

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

System Message: WARNING/2 ( h\left[X\right]=\log\left(2\pi\right).)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\log\left(2\pi\right). (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 284 bytes). Transcript written on math.log.

Gauss Hypergeometric¶

$x\in\left[0,1\right]$ , $\alpha>0,\,\beta>0$

System Message: WARNING/2 ( C^{-1}=B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 C^ {-1}=B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alph... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 492 bytes). Transcript written on math.log.

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

Inverted Gamma¶

Special case of the generalized Gamma distribution with $c=-1$ and $a>0$ , $x>0$

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 460 bytes). Transcript written on math.log.

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

System Message: WARNING/2 ( m_{d}=\frac{1}{a+1})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 m_ {d}=\frac{1}{a+1} (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 320 bytes). Transcript written on math.log.

System Message: WARNING/2 ( h\left[X\right]=a-\left(a+1\right)\Psi\left(a\right)+\log\Gamma\left(a\right).)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=a-\left(a+1\right)\Psi\left(a\right)+\log\Gamma\left(... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 388 bytes). Transcript written on math.log.

Inverse Normal (Inverse Gaussian)¶

The standard form involves the shape parameter $\mu$ (in most definitions, $L=0.0$ is used). (In terms of the regress documentation $\mu=A/B$ ) and $B=S$ and $L$ is not a parameter in that distribution. A standard form is $x>0$

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

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

This is related to the canonical form or JKB “two-parameter “inverse Gaussian when written in it’s full form with scale parameter $S$ and location parameter $L$ by taking $L=0$ and $S\equiv\lambda,$ then $\mu S$ is equal to $\mu_{2}$ where $\mu_{2}$ 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 $\left(\sqrt{\beta_{1}}\right)$ and the kurtosis ( $\beta_{2}-3$ ) are both functions only of $\mu_{2}/\lambda=\mu S/S=\mu$ as shown here, while the variance and mean of the standard form here are transformed appropriately.

Inverted Weibull¶

Shape parameter $c>0$ and $x>0$ . Then

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

System Message: WARNING/2 ( h\left[X\right]=1+\gamma+\frac{\gamma}{c}-\log\left(c\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=1+\gamma+\frac{\gamma}{c}-\log\left(c\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 428 bytes). Transcript written on math.log.

where $\gamma$ is Euler’s constant.

Johnson SB¶

Defined for $x\in\left(0,1\right)$ with two shape parameters $a$ and $b>0.$

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

Johnson SU¶

Defined for all $x$ with two shape parameters $a$ and $b>0$ .

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

KSone¶

KStwo¶

Laplace (Double Exponential, Bilateral Expoooonential)¶

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

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

The ML estimator of the location parameter is

System Message: WARNING/2 ( \hat{L}=\textrm{median}\left(X_{i}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Package amsmath Error: \hat allowed only in math mode. See the amsmath package documentation for explanation. Type H <return> for immediate help. ... l.14 \hat{L} =\textrm{median}\left(X_{i}\right) ! Missing $ inserted. <inserted text> $ l.14 \hat{L}=\textrm{median}\left (X_{i}\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 296 bytes). Transcript written on math.log.

where $X_{i}$ is a sequence of $N$ mutually independent Laplace RV’s and the median is some number between the $\frac{1}{2}N\textrm{th}$ and the $(N/2+1)\textrm{th}$ order statistic ( e.g. take the average of these two) when $N$ is even. Also,

System Message: WARNING/2 ( \hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|.)

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Replace $\hat{L}$ with $L$ if it is known. If $L$ is known then this estimator is distributed as $\left(2N\right)^{-1}S\cdot\chi_{2N}^{2}$ .

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

Left-skewed Lévy¶

Special case of Lévy-stable distribution with $\alpha=\frac{1}{2}$ and $\beta=-1$ the support is $x<0$ . In standard form

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

No moments.

Lévy¶

A special case of Lévy-stable distributions with $\alpha=\frac{1}{2}$ and $\beta=1$ . In standard form it is defined for $x>0$ as

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

It has no finite moments.

Logistic (Sech-squared)¶

A special case of the Generalized Logistic distribution with $c=1.$ Defined for $x>0$

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

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

System Message: WARNING/2 ( h\left[X\right]=1.)

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Log Double Exponential (Log-Laplace)¶

Defined over $x>0$ with $c>0$

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

System Message: WARNING/2 ( h\left[X\right]=\log\left(\frac{2e}{c}\right))

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Log Gamma¶

A single shape parameter $c>0$ (Defined for all $x$ )

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\int_{0}^{\infty}\left[\log y\right]^{n}y^{c-1}\exp\left(-y\right)dy.)

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

Log Normal (Cobb-Douglass)¶

Has one shape parameter $\sigma$ >0. (Notice that the “Regress “ $A=\log S$ where $S$ is the scale parameter and $A$ is the mean of the underlying normal distribution). The standard form is $x>0$

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

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

Notice that using JKB notation we have $\theta=L,$ $\zeta=\log S$ 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.

System Message: WARNING/2 ( h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right].)

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Also, note that if $X$ is a log-normally distributed random-variable with $L=0$ and $S$ and shape parameter $\sigma.$ Then, $\log X$ is normally distributed with variance $\sigma^{2}$ and mean $\log S.$

Nakagami¶

Generalization of the chi distribution. Shape parameter is $\nu>0.$ Defined for $x>0.$

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

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

Noncentral beta*¶

Defined over $x\in\left[0,1\right]$ with $a>0$ and $b>0$ and $c\geq0$

System Message: WARNING/2 ( 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))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 F\left (x;a,b,c\right)=\sum_{j=0}^{\infty}\frac{e^{-c/2}\left(\frac{c}... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 792 bytes). Transcript written on math.log.

Noncentral chi*¶

Noncentral chi-squared¶

The distribution of $\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}$ where $Z_{i}$ are independent standard normal variables and $\delta_{i}$ are constants. $\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.$ (In communications it is called the Marcum-Q function). Can be thought of as a Generalized Rayleigh-Rice distribution. For $x>0$

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

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

Noncentral F¶

Let $\lambda>0$ and $\nu_{1}>0$ and $\nu_{2}>0.$

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

Noncentral t¶

The distribution of the ratio

System Message: WARNING/2 ( \frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}})

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where $U$ and $\chi_{\nu}$ are independent and distributed as a standard normal and chi with $\nu$ degrees of freedom. Note $\lambda>0$ and $\nu>0$ .

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

Normal¶

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

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

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

Maxwell¶

This is a special case of the Chi distribution with $L=0$ and $S=S=\frac{1}{\sqrt{a}}$ and $\nu=3.$

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

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

System Message: WARNING/2 ( h\left[X\right]=\log\left(\sqrt{\frac{2\pi}{e}}\right)+\gamma.)

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Mielke’s Beta-Kappa¶

A generalized F distribution. Two shape parameters $\kappa$ and $\theta$ , and $x>0$ . The $\beta$ in the DATAPLOT reference is a scale parameter.

$\begin{eqnarray*} f\left(x;\kappa,\theta\right) & = & \frac{\kappa x^{\kappa-1}}{\left(1+x^{\theta}\right)^{1+\frac{\kappa}{\theta}}}\\ F\left(x;\kappa,\theta\right) & = & \frac{x^{\kappa}}{\left(1+x^{\theta}\right)^{\kappa/\theta}}\\ G\left(q;\kappa,\theta\right) & = & \left(\frac{q^{\theta/\kappa}}{1-q^{\theta/\kappa}}\right)^{1/\theta}\end{eqnarray*}$

Pareto¶

For $x\geq1$ and $b>0$ . Standard form is

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

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

System Message: WARNING/2 ( h\left(X\right)=\frac{1}{c}+1-\log\left(c\right))

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Pareto Second Kind (Lomax)¶

$c>0.$ This is Pareto of the first kind with $L=-1.0$ so $x\geq0$

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

System Message: WARNING/2 ( h\left[X\right]=\frac{1}{c}+1-\log\left(c\right).)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\frac{1}{c}+1-\log\left(c\right). (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 464 bytes). Transcript written on math.log.

Power Log Normal¶

A generalization of the log-normal distribution $\sigma>0$ and $c>0$ and $x>0$

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\int_{0}^{1}\exp\left[-n\sigma\Phi^{-1}\left(y^{1/c}\right)\right]dy)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\int_{0}^{1}\exp\left[-n\sigma\Phi^{-1}\left(y^{1/c}... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 604 bytes). Transcript written on math.log.

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

This distribution reduces to the log-normal distribution when $c=1.$

Power Normal¶

A generalization of the normal distribution, $c>0$ for

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\left(-1\right)^{n}\int_{0}^{1}\left[\Phi^{-1}\left(y^{1/c}\right)\right]^{n}dy)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\left(-1\right)^{n}\int_{0}^{1}\left[\Phi^{-1}\left(... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 632 bytes). Transcript written on math.log.

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

For $c=1$ this reduces to the normal distribution.

Power-function¶

A special case of the beta distribution with $b=1$ : defined for $x\in\left[0,1\right]$

$a>0$

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

System Message: WARNING/2 ( h\left[X\right]=1-\frac{1}{a}-\log\left(a\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=1-\frac{1}{a}-\log\left(a\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 456 bytes). Transcript written on math.log.

R-distribution¶

A general-purpose distribution with a variety of shapes controlled by $c>0.$ Range of standard distribution is $x\in\left[-1,1\right]$

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

System Message: WARNING/2 ( \mu_{n}^{\prime}=\frac{\left(1+\left(-1\right)^{n}\right)}{2}B\left(\frac{n+1}{2},\frac{c}{2}\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\frac{\left(1+\left(-1\right)^{n}\right)}{2}B\left(\... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 676 bytes). Transcript written on math.log.

The R-distribution with parameter $n$ is the distribution of the correlation coefficient of a random sample of size $n$ drawn from a bivariate normal distribution with $\rho=0.$ 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.

Rayleigh¶

This is Chi distribution with $L=0.0$ and $\nu=2$ and $S=S$ (no location parameter is generally used), the mode of the distribution is $S.$

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

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

System Message: WARNING/2 ( h\left[X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right).)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right). (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 580 bytes). Transcript written on math.log.

System Message: WARNING/2 ( \mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 592 bytes). Transcript written on math.log.

Rice*¶

Defined for $x>0$ and $b>0$

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

System Message: WARNING/2 ( \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))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(1+\frac{n}{2}\right)\,_{1}F_... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 808 bytes). Transcript written on math.log.

Reciprocal¶

Shape parameters $a,b>0$ $x\in\left[a,b\right]$

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

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

System Message: WARNING/2 ( h\left[X\right]=\frac{1}{2}\log\left(ab\right)+\log\left[\log\left(\frac{b}{a}\right)\right].)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\frac{1}{2}\log\left(ab\right)+\log\left[\log\left(\f... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 564 bytes). Transcript written on math.log.

Reciprocal Inverse Gaussian¶

The pdf is found from the inverse gaussian (IG), $f_{RIG}\left(x;\mu\right)=\frac{1}{x^{2}}f_{IG}\left(\frac{1}{x};\mu\right)$ defined for $x\geq0$ as

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

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

Semicircular¶

Defined on $x\in\left[-1,1\right]$

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

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

System Message: WARNING/2 ( h\left[X\right]=0.64472988584940017414.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=0.64472988584940017414. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 276 bytes). Transcript written on math.log.

Studentized Range*¶

Student t¶

Shape parameter $\nu>0.$ $I\left(a,b,x\right)$ is the incomplete beta integral and $I^{-1}\left(a,b,I\left(a,b,x\right)\right)=x$

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

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

As $\nu\rightarrow\infty,$ this distribution approaches the standard normal distribution.

System Message: WARNING/2 ( 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])

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\frac{1}{4}\log\left(\frac{\pi c\Gamma^{2}\left(\frac... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 1104 bytes). Transcript written on math.log.

where

System Message: WARNING/2 ( 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)})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 Z\left (c\right)=\,_{3}F_{2}\left(1,1,1+\frac{c}{2};\frac{3}{2},2;1\ri... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 1136 bytes). Transcript written on math.log.

Student Z¶

The student Z distriubtion is defined over all space with one shape parameter $\nu>0$

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

Interesting moments are

$\begin{eqnarray*} \mu & = & 0\\ \sigma^{2} & = & \frac{1}{\nu-3}\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & \frac{6}{\nu-5}.\end{eqnarray*}$

The moment generating function is

System Message: WARNING/2 ( \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)}.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \theta \left(t\right)=2\sqrt{\left|\frac{t}{2}\right|^{\nu-1}}\frac{K_... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 888 bytes). Transcript written on math.log.

Symmetric Power*¶

Triangular¶

One shape parameter $c\in[0,1]$ 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-parameter is the width of the non-zero portion. In standard form we have $x\in\left[0,1\right].$

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

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

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

Truncated Exponential¶

This is an exponential distribution defined only over a certain region $0<x<B$ . In standard form this is

$\begin{eqnarray*} f\left(x;B\right) & = & \frac{e^{-x}}{1-e^{-B}}\\ F\left(x;B\right) & = & \frac{1-e^{-x}}{1-e^{-B}}\\ G\left(q;B\right) & = & -\log\left(1-q+qe^{-B}\right)\end{eqnarray*}$

System Message: WARNING/2 ( \mu_{n}^{\prime}=\Gamma\left(1+n\right)-\Gamma\left(1+n,B\right))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 \mu _{n}^{\prime}=\Gamma\left(1+n\right)-\Gamma\left(1+n,B\right) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 448 bytes). Transcript written on math.log.

System Message: WARNING/2 ( h\left[X\right]=\log\left(e^{B}-1\right)+\frac{1+e^{B}\left(B-1\right)}{1-e^{B}}.)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo aded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texmf/tex/latex/preview/preview.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) Preview: Fontsize 12pt ! Missing $ inserted. <inserted text> $ l.14 h\left [X\right]=\log\left(e^{B}-1\right)+\frac{1+e^{B}\left(B-1\right... (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing $ inserted. <inserted text> $ l.15 \end{preview} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 664 bytes). Transcript written on math.log.

Truncated Normal¶

A normal distribution restricted to lie within a certain range given by two parameters $A$ and $B$ . Notice that this $A$ and $B$ correspond to the bounds on $x$ in standard form. For $x\in\left[A,B\right]$ we get

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

where

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

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

Tukey-Lambda¶

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

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

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

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

Uniform¶

Standard form $x\in\left(0,1\right).$ In general form, the lower limit is $L,$ the upper limit is $S+L.$

$\begin{eqnarray*} f\left(x\right) & = & 1\\ F\left(x\right) & = & x\\ G\left(q\right) & = & q\end{eqnarray*}$

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

System Message: WARNING/2 ( h\left[X\right]=0)

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Von Mises¶

Defined for $x\in\left[-\pi,\pi\right]$ with shape parameter $b>0$ . Note, the PDF and CDF functions are periodic and are always defined over $x\in\left[-\pi,\pi\right]$ 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 $b<100$ the PDF and CDF formulas below are used. Otherwise, a normal approximation with variance $1/b$ is used.

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

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

This can be used for defining circular variance.

Wald¶

Special case of the Inverse Normal with shape parameter set to $1.0$ . Defined for $x>0$ .

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

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

Wishart*¶

Wrapped Cauchy¶

For $x\in\left[0,2\pi\right]$ $c\in\left(0,1\right)$

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

System Message: WARNING/2 ( h\left[X\right]=\log\left(2\pi\left(1-c^{2}\right)\right).)

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Navigation

Continuous Statistical Distributions¶

Overview¶

Moments¶

Median and mode¶

Fitting data¶

Standard notation for mean¶

References¶

Alpha¶

Anglit¶

Arcsine¶

Beta¶

Beta Prime¶

Bradford¶

Burr¶

Cauchy¶

Chi¶

Chi-squared¶

Cosine¶

Double Gamma¶

Doubly Non-central F*¶

Doubly Non-central t*¶

Double Weibull¶

Erlang¶

Exponential¶

Exponentiated Weibull¶

Exponential Power¶

Fatigue Life (Birnbaum-Sanders)¶

Fisk (Log Logistic)¶

Folded Cauchy¶

Folded Normal¶

Fratio (or F)¶

Fréchet (ExtremeLB, Extreme Value II, Weibull minimum)¶

Fréchet (left-skewed, Extreme Value Type III, Weibull maximum)¶

Gamma¶

Generalized Logistic¶

Generalized Pareto¶

Generalized Exponential¶

Generalized Extreme Value¶

Generalized Gamma¶

Generalized Half-Logistic¶

Gilbrat¶

Gompertz (Truncated Gumbel)¶

Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value)¶

Gumbel Left-skewed (for minimum order statistic)¶

HalfCauchy¶

HalfNormal¶

Half-Logistic¶

Hyperbolic Secant¶

Gauss Hypergeometric¶

Inverted Gamma¶

Inverse Normal (Inverse Gaussian)¶

Inverted Weibull¶

Johnson SB¶

Johnson SU¶

KSone¶

KStwo¶

Laplace (Double Exponential, Bilateral Expoooonential)¶

Left-skewed Lévy¶

Lévy¶

Logistic (Sech-squared)¶

Log Double Exponential (Log-Laplace)¶

Log Gamma¶

Log Normal (Cobb-Douglass)¶

Nakagami¶

Noncentral beta*¶

Noncentral chi*¶

Noncentral chi-squared¶

Noncentral F¶

Noncentral t¶

Normal¶

Maxwell¶

Mielke’s Beta-Kappa¶

Pareto¶

Pareto Second Kind (Lomax)¶

Power Log Normal¶

Power Normal¶

Power-function¶

R-distribution¶

Rayleigh¶