### Navigation

- index
- modules |
- modules |
- SciPy v0.12 Reference Guide (DRAFT) »

# 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 and The nonstandard forms can be obtained for the various functions using (note is a standard uniform random variate).

Function Name | Standard Function | Transformation
--- | --- | ---
Cumulative Distribution Function (CDF) |  |
Probability Density Function (PDF) |  |
Percent Point Function (PPF) |  |
Probability Sparsity Function (PSF) |  |
Hazard Function (HF) |  |
Cumulative Hazard Functon (CHF) |  |
Survival Function (SF) |  |
Inverse Survival Function (ISF) |  |
Moment Generating Function (MGF) |  |
Random Variates |  |
(Differential) Entropy |  |
(Non-central) Moments |  |
Central Moments |  |
mean (mode, median), var |  |
skewness, kurtosis |  |

### Moments

Non-central moments are defined using the PDF

System Message: WARNING/2 (`\mu_{n}^{\prime}=\int_{-\infty}^{\infty}x^{n}f\left(x\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 \mu _{n}^{\prime}=\int_{-\infty}^{\infty}x^{n}f\left(x\right)dx. (/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.

Note, that these can always be computed using the PPF. Substitute in the above equation and get

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

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}G^{n}\left(q\right)dq (/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.

which may be easier to compute numerically. Note that so that Central moments are computed similarly

In particular

Skewness is defined as

System Message: WARNING/2 (`\gamma_{1}=\sqrt{\beta_{1}}=\frac{\mu_{3}}{\mu_{2}^{3/2}}`)

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 \gamma _{1}=\sqrt{\beta_{1}}=\frac{\mu_{3}}{\mu_{2}^{3/2}} (/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, 568 bytes). Transcript written on math.log.

while (Fisher) kurtosis is

System Message: WARNING/2 (`\gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,`)

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 _{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3, (/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, 484 bytes). Transcript written on math.log.

so that a normal distribution has a kurtosis of zero.

### Median and mode

The median, is defined as the point at which half of the density is on one side and half on the other. In other words, so that

System Message: WARNING/2 (`m_{n}=G\left(\frac{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 m_ {n}=G\left(\frac{1}{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, 440 bytes). Transcript written on math.log.

In addition, the mode, , is defined as the value for which the probability density function reaches it’s peak

System Message: WARNING/2 (`m_{d}=\arg\max_{x}f\left(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 m_ {d}=\arg\max_{x}f\left(x\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, 340 bytes). Transcript written on math.log.

### 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 is the PDF of a random-variable where is a vector of parameters ( e.g. and ), then for a collection of independent samples from this distribution, the joint distribution the random vector is

System Message: WARNING/2 (`f\left(\mathbf{x};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f\left(x_{i};\boldsymbol{\theta}\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 (\mathbf{x};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f\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, 564 bytes). Transcript written on math.log.

The maximum likelihood estimate of the parameters are the parameters which maximize this function with fixed and given by the data:

Where

Note that if includes only shape parameters, the location and scale-parameters can be fit by replacing with in the log-likelihood function adding and minimizing, thus

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

where and are assumed known as the mean and variance of the untransformed distribution (when and ) and

### 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 should be clear from context as the number of samples

### 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 (paramter in DATAPLOT is a scale-parameter). Standard form is

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

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 . To get the JKB definition put i.e. and

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.

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

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

## Beta Prime

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

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,

## Bradford

where 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

## Cauchy

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

## Chi

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

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)`)

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(\frac{v}{2}\right)\,_{1}F_{1}\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, 1108 bytes). Transcript written on math.log.

## Chi-squared

This is the gamma distribution with and and where is called the degrees of freedom. If are all standard normal distributions, then has (standard) chi-square distribution with degrees of freedom.

The standard form (most often used in standard form only) is

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.

## Cosine

Approximation to the normal distribution.

## Double Gamma

The double gamma is the signed version of the Gamma distribution. For

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.

## Doubly Non-central F*

## Doubly Non-central t*

## Double Weibull

This is a signed form of the Weibull distribution.

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.

## Erlang

This is just the Gamma distribution with shape parameter an integer.

## Exponential

This is a special case of the Gamma (and Erlang) distributions with shape parameter and the same location and scale parameters. The standard form is therefore ( )

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.

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 and and

## Exponential Power

One positive shape parameter . Defined for

## Fatigue Life (Birnbaum-Sanders)

This distribution’s pdf is the average of the inverse-Gaussian and reciprocal inverse-Gaussian pdf . We follow the notation of JKB here with for

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.

## Fisk (Log Logistic)

Special case of the Burr distribution with

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 and the pdf ). if is cauchy then is folded cauchy. Note that

No moments

## Folded Normal

If is Normal with mean and , then is a folded normal with shape parameter , location parameter and scale parameter . This is a special case of the non-central chi distribution with one- degree of freedom and non-centrality parameter Note that . The standard form of the folded normal is

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.

## Fratio (or F)

Defined for . The distribution of if is chi-squared with degrees of freedom and is chi-squared with degrees of freedom.

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

A type of extreme-value distribution with a lower bound. Defined for and

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.

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 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 and .

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 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 valid for .

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.

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 And

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.

Note that the polygamma function is

where is a generalization of the Riemann zeta function called the Hurwitz zeta function Note that

## Generalized Pareto

Shape parameter and defined for for all and if is negative.

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.

Thus,

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 Note that and are all

## Generalized Extreme Value

Extreme value distributions with shape parameter .

For defined on

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,

For defined on For defined over all space

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

## Generalized Gamma

A general probability form that reduces to many common distributions: and

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.

Special cases are Weibull , half-normal and ordinary gamma distributions If 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 and we have

## Gilbrat

Special case of the log-normal with and (typically also )

## Gompertz (Truncated Gumbel)

For and . In JKB the two shape parameters are reduced to the single shape-parameter . As is just a scale parameter when . If the distribution reduces to the exponential distribution scaled by Thus, the standard form is given as

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).

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.

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)

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 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 is Hyperbolic Secant distributed then is Half-Cauchy distributed. Also, if is (standard) Cauchy distributed, then is Half-Cauchy distributed. Special case of the Folded Cauchy distribution with The standard form is

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.

No moments, as the integrals diverge.

## HalfNormal

This is a special case of the chi distribution with and and This is also a special case of the folded normal with shape parameter and If is (standard) normally distributed then, is half-normal. The standard form is

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.

## Half-Logistic

In the limit as for the generalized half-logistic we have the half-logistic defined over Also, the distribution of where has logistic distribtution.

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.

## Hyperbolic Secant

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

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.

where is an integer given by

where is the Bernoulli polynomial of order evaluated at 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.

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

,

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.

## Inverted Gamma

Special case of the generalized Gamma distribution with and ,

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.

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 (in most definitions, is used). (In terms of the regress documentation ) and and is not a parameter in that distribution. A standard form is

This is related to the canonical form or JKB “two-parameter “inverse Gaussian when written in it’s full form with scale parameter and location parameter by taking and then is equal to where 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 and the kurtosis ( ) are both functions only of as shown here, while the variance and mean of the standard form here are transformed appropriately.

## Inverted Weibull

Shape parameter and . Then

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 is Euler’s constant.

## Johnson SB

Defined for with two shape parameters and

## Johnson SU

Defined for all with two shape parameters and .

## KSone

## KStwo

## Laplace (Double Exponential, Bilateral Expoooonential)

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 is a sequence of mutually independent Laplace RV’s and the median is some number between the and the order statistic ( e.g. take the average of these two) when is even. Also,

System Message: WARNING/2 (`\hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\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{S} =\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|. ! Missing $ inserted. <inserted text> $ l.14 \hat{S}=\frac{1}{N} \sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \@@over #2} l.14 \hat{S}=\frac{1}{N} \sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|. ! Missing $ inserted. <inserted text> $ l.15 \end{preview} ! 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, 616 bytes). Transcript written on math.log.

Replace with if it is known. If is known then this estimator is distributed as .

## Left-skewed Lévy

Special case of Lévy-stable distribution with and the support is . In standard form

No moments.

## Lévy

A special case of Lévy-stable distributions with and . In standard form it is defined for as

It has no finite moments.

## Logistic (Sech-squared)

A special case of the Generalized Logistic distribution with Defined for

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.

## Log Double Exponential (Log-Laplace)

Defined over with

System Message: WARNING/2 (`h\left[X\right]=\log\left(\frac{2e}{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]=\log\left(\frac{2e}{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, 452 bytes). Transcript written on math.log.

## Log Gamma

A single shape parameter (Defined for all )

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

## Log Normal (Cobb-Douglass)

Has one shape parameter >0. (Notice that the “Regress “ where is the scale parameter and is the mean of the underlying normal distribution). The standard form is

Notice that using JKB notation we have 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].`)

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}\left[1+\log\left(2\pi\right)+2\log\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, 428 bytes). Transcript written on math.log.

Also, note that if is a log-normally distributed random-variable with and and shape parameter Then, is normally distributed with variance and mean

## Nakagami

Generalization of the chi distribution. Shape parameter is Defined for

## Noncentral beta*

Defined over with and and

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 where are independent standard normal variables and are constants. (In communications it is called the Marcum-Q function). Can be thought of as a Generalized Rayleigh-Rice distribution. For

## Noncentral F

Let and and

## Noncentral t

The distribution of the ratio

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

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 \frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}} (/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.14 \frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}} ! Missing $ inserted. <inserted text> $ l.14 \frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}} ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \@@over #2} l.14 \frac{U+\lambda}{\chi_{\nu}/\sqrt{\nu}} ! Missing $ inserted. <inserted text> $ l.15 \end{preview} ! 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 and are independent and distributed as a standard normal and chi with degrees of freedom. Note and .

## Normal

## Maxwell

This is a special case of the Chi distribution with and and

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

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(\sqrt{\frac{2\pi}{e}}\right)+\gamma. (/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.

## Mielke’s Beta-Kappa

A generalized F distribution. Two shape parameters and , and . The in the DATAPLOT reference is a scale parameter.

## Pareto

For and . Standard form is

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, 456 bytes). Transcript written on math.log.

## Pareto Second Kind (Lomax)

This is Pareto of the first kind with so

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 and and

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.

This distribution reduces to the log-normal distribution when

## Power Normal

A generalization of the normal distribution, for

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.

For this reduces to the normal distribution.

## Power-function

A special case of the beta distribution with : defined for

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 Range of standard distribution is

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 is the distribution of the correlation coefficient of a random sample of size drawn from a bivariate normal distribution with 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 and and (no location parameter is generally used), the mode of the distribution is

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 and

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

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), defined for as

## Semicircular

Defined on

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 is the incomplete beta integral and

As 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

Interesting moments are

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 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

## Truncated Exponential

This is an exponential distribution defined only over a certain region . In standard form this is

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 and . Notice that this and correspond to the bounds on in standard form. For we get

where

## Tukey-Lambda

Notice that the

## Uniform

Standard form In general form, the lower limit is the upper limit is

System Message: WARNING/2 (`h\left[X\right]=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]=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, 248 bytes). Transcript written on math.log.

## Von Mises

Defined for with shape parameter . Note, the PDF and CDF functions are periodic and are always defined over 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 the PDF and CDF formulas below are used. Otherwise, a normal approximation with variance is used.

This can be used for defining circular variance.

## Wald

Special case of the Inverse Normal with shape parameter set to . Defined for .

## Wishart*

## Wrapped Cauchy

For

System Message: WARNING/2 (`h\left[X\right]=\log\left(2\pi\left(1-c^{2}\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]=\log\left(2\pi\left(1-c^{2}\right)\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, 408 bytes). Transcript written on math.log.

### Navigation

- index
- modules |
- modules |
- SciPy v0.12 Reference Guide (DRAFT) »