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Diffstat (limited to 'Master/texmf-dist/doc/generic/pst-func/pst-func-doc.tex')
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diff --git a/Master/texmf-dist/doc/generic/pst-func/pst-func-doc.tex b/Master/texmf-dist/doc/generic/pst-func/pst-func-doc.tex index 880c2649723..d08f8bedaad 100644 --- a/Master/texmf-dist/doc/generic/pst-func/pst-func-doc.tex +++ b/Master/texmf-dist/doc/generic/pst-func/pst-func-doc.tex @@ -1,4 +1,4 @@ -%% $Id: pst-func-doc.tex 273 2010-01-26 18:28:55Z herbert $ +%% $Id: pst-func-doc.tex 285 2010-02-11 09:40:27Z herbert $ \documentclass[11pt,english,BCOR10mm,DIV12,bibliography=totoc,parskip=false, smallheadings, headexclude,footexclude,oneside]{pst-doc} \usepackage[utf8]{inputenc} @@ -1403,6 +1403,74 @@ and has the syntax (with a default setting of $s=1$ and $\mu=1$): \clearpage +\subsection{Weibull distribution} + +In probability theory and statistics, the Weibull distribution is a continuous probability +distribution. The probability density function of a +Weibull random variable $x$ is: + +\begin{align} +P(x) &= \alpha\beta^{-\alpha} x^{\alpha-1} e^{-\left(\frac{x}{\beta}\right)^\alpha}\\ +D(x) &= 1-e^{-\left(\frac{x}{\beta}\right)^\alpha} +\end{align} + +or slightly different as + +\begin{align} +P(x) &= \frac{\alpha}{\beta}\,x^{\alpha-1} e^{-\frac{x^\alpha}{\beta}}\\ +D(x) &= 1 - e^{-\frac{x^\alpha}{\beta}} +\end{align} + +always for $x\in[0;\infty)$. +where $\alpha > 0$ is the shape parameter and $\beta > 0$ is the scale parameter of the distribution. + +$D(x)$ is the cumulative distribution function of the Weibull distribution. The values for +$\alpha$ and $\beta$ are preset to 1, but can be changed in the usual way. + +The Weibull distribution is related to a number of other probability distributions; in +particular, it interpolates between the exponential distribution $(\alpha = 1)$ and the +Rayleigh distribution $(\alpha = 2)$. + +\begin{center} +\psset{unit=2} +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5}{5}{% + \psWeibull[alpha=\rAlpha]{0}{2.5} + \psWeibullI[alpha=\rAlpha,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +% +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5,\rBeta=0.2+0.2}{5}{% + \psWeibull[alpha=\rAlpha,beta=\rBeta]{0}{2.5} + \psWeibullI[alpha=\rAlpha,beta=\rBeta,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +\end{center} + +\begin{lstlisting} +\psset{unit=2} +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5}{5}{% + \psWeibull[alpha=\rAlpha]{0}{2.5} + \psWeibullI[alpha=\rAlpha,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +% +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5,\rBeta=0.2+0.2}{5}{% + \psWeibull[alpha=\rAlpha,beta=\rBeta]{0}{2.5} + \psWeibullI[alpha=\rAlpha,beta=\rBeta,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +\end{lstlisting} +\psset{unit=1cm} + +The starting value for $x$ should always be 0 or greater, if it is +less than 0 then the macro draws a line from (\#1,0) to (0,0) and +starts \Lcs{psWeinbull} with 0. + +\clearpage \section{The Lorenz curve} The so-called \Index{Lorenz curve} is used in economics to describe inequality in |