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authorNorbert Preining <norbert@preining.info>2023-04-15 03:01:57 +0000
committerNorbert Preining <norbert@preining.info>2023-04-15 03:01:57 +0000
commit78e6b19d4678a9d641c4afa4ca800288cfa0a9d5 (patch)
treead774593055c02ade73402379b9df52568f6940f /graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex
parentb17717e3829bc08efb71ec2537d9d894c8b82d45 (diff)
CTAN sync 202304150301
Diffstat (limited to 'graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex')
-rw-r--r--graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex110
1 files changed, 108 insertions, 2 deletions
diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex
index b3f0c2ecae..8180a0e80e 100644
--- a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex
+++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex
@@ -1,6 +1,6 @@
\RequirePackage{pdfmanagement-testphase}
\DeclareDocumentMetadata{}
-%% $Id: pst-func-doc.tex 691 2023-04-03 10:51:04Z herbert $
+%% $Id: pst-func-doc.tex 697 2023-04-14 10:17:32Z herbert $
\documentclass[fontsize=11pt,english,BCOR=10mm,DIV=12,bibliography=totoc,parskip=false,headings=small,
headinclude=false,footinclude=false,oneside]{pst-doc}
\usepackage{pst-func}
@@ -1435,7 +1435,8 @@ and is predefined with 5.
\subsection{Binomial distribution}\label{sec:bindistri}
\begin{sloppypar}
-The following five macros plot binomial probability mass function \Lcs{psBinomial} and \Lcs{psBinomialC} in curve style, the normalized one is \Lcs{psBinomialN}. The cumulative distribution function $F$ \Lcs{psBinomialF} and the complement of the cumulative distribution function ($1-F$) \Lcs{psBinomialFS}
+The following five macros plot binomial probability mass function \Lcs{psBinomial} and \Lcs{psBinomialC} in curve style, the normalized one is \Lcs{psBinomialN}. The cumulative distribution function
+$F$ \Lcs{psBinomialF} and the complement of the cumulative distribution function ($1-F$) \Lcs{psBinomialFS}
The vertical range for the plots is the $y$-Intervall $[0;1]$.
Rescaling other values can be done by setting the \Lkeyword{yunit} option
to any other value.
@@ -2008,6 +2009,111 @@ The default settings are $\mu=1$ and $\nu=1$.
\end{LTXexample}
\clearpage
+
+\subsection{Noncentral $F$-distribution}
+
+The noncentral F-distribution is given by
+
+
+\begin{align}
+P(x)=\sum\limits_{k=0}^{\infty}{\frac{e^{-\lambda/2}(\lambda/2)^{k}}
+ {B\left({\frac {\nu_{2}}{2}},{\frac {\nu _{1}}{2}}+k\right)k!}}
+\left({\frac {\nu _{1}}{\nu _{2}}}\right)^{{\frac {\nu _{1}}{2}}+k}
+ \left({\frac {\nu _{2}}{\nu_{2}
+ +\nu _{1}f}}\right)^{{\frac{\nu_{1}
+ +\nu _{2}}{2}}+k}x^{\nu_{1}/2-1+k}
+\end{align}
+
+
+\iffalse
+\begin{align}
+P(x) &= e^\frac{\frac{-\lambda}2+(\lambda n_1x)}{2(n_2+n_1x)}
+n_1^{\frac{n_1}2} n_2^{\frac{n_2}2} x^{\frac{n_1}2-1}
+(n_2+n_1x)^{\frac{-(n_1+n_2)}2}
+\frac{\Gamma(\frac12 n_1)\Gamma\left(1+\frac12 n_2\right)
+ \mathop{L}_\frac{n_2}2^{\frac{n_1}2-1}\left(-\frac{\lambda n_1 x}{2(n_2+n_1x)}\right)}
+ {B\left(\frac12 n_1,\frac12 n_2\right)\Gamma\left(\frac12 (n_1+n_2)\right)}
+\\
+ &= \frac1{B\left(\frac12 n_1,\frac12 n_2\right)}
+ \left(
+ e^{\frac\lambda2}n_1^{\frac{n_1}2} n_2^{\frac{n_2}2} x^{\frac{n_1}2-1}
+ (n_2+n_1x)^{\frac{-(n_1+n_2)}2}
+ {}_1F_1\left( \frac12(n_1+n_2); \, \frac12n_1; \, \frac{\lambda n_1 x}{2(n_2+n_1x)} \right)
+ \right)
+\end{align}
+
+
+%(e^(lambda/2)x^(n_1/2-1)(xn_1+n_2)^(-(n_1+n_2)/2)n_1^(n_1/2)n_2^(n_2/2)_1F_1(1/2(n_1+n_2);1/2n_1;(xlambdan_1)/(2(xn_1+n_2)))),
+%\]
+
+\fi
+
+
+where %$\Gamma(z)$ is the gamma function,
+$B(\alpha,\beta)$ is the beta function
+
+\begin{align}
+B(p,q) &= \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}\\
+ &= \frac{(p-1)!(q-1)!}{(p+q-1)!}
+\end{align}
+
+\iffalse
+ and $L_m^n(z)$ is a generalized Laguerre polynomial.
+
+ \begin{align}
+% L_n(x) &= \sum_{k=0}^n \frac{(-1)^k}{k!} \binom{n}{k} x^k \\
+L_n^{k(x)} &= \frac{e^xx^{-k}}{n!} \frac{d^n}{dx^n} e^{-x}x^{n+k} \\
+ &= (-1)^k \frac{d^k}{dx^k} L_{n+k}(x)\\
+ &= \frac{(-1)^nx^(-(k+1)/2)}{n!} e^\frac x2 W_{\frac k2+n+\frac12, \frac k2}(x)\\
+ &= \sum_{m=0}^n(-1)^m \frac{(n+k)!}{(n-m)!(k+m)!m!}x^m
+ \end{align}
+
+where $W_{k,m}(x)$ is a Whittaker function.
+
+\fi
+
+
+
+\begin{BDef}
+\Lcs{psNonCentralFDist}\OptArgs\Largb{x0}\Largb{x1}
+\end{BDef}
+%
+The default settings are $\mu=1$, $\nu=1$ and $\lambda=0$, which is the same as the default F-distribution.
+
+\begin{LTXexample}[pos=t,preset=\centering]
+\psset{xunit=1cm,yunit=10cm}
+\begin{pspicture*}(-2,-0.02)(10.5,1.1)
+\psset{linewidth=1pt}
+% \psFDist[linecolor=blue,mue=5,nue=20,lambda=0]{0.01}{5}
+\psaxes[Dy=0.1]{->}(0,0)(10,1)
+\psFDist[linecolor=red,mue=5,nue=20]{0.01}{10}
+\psNonCentralFDist[linecolor=green,mue=5,nue=20,lambda=1.6]{0.01}{10}
+\end{pspicture*}
+\end{LTXexample}
+
+
+\begin{LTXexample}[pos=t,preset=\centering]
+\psset{xunit=2cm,yunit=10cm,plotpoints=100}
+\begin{pspicture*}(-0.5,-0.07)(5.5,0.8)
+ \psline[linestyle=dashed](0.5,0)(0.5,0.75)
+ \psline[linestyle=dashed](! 2 7 div 0)(! 2 7 div 0.75)
+ \psset{linewidth=1pt}
+ \psFDist{0.1}{5}
+ \psFDist[linecolor=red,mue=5,nue=20]{0.01}{5}
+ \psNonCentralFDist[linecolor=green,mue=5,nue=20,lambda=1.6]{0.01}{5}
+ \psNonCentralFDist[linecolor=blue,nue=5,mue=20,lambda=1.6]{0.01}{5}
+ \psFDist[linecolor=blue,mue=12,nue=3]{0.01}{5}
+ \psNonCentralFDist[linecolor=green,mue=12,nue=3,lambda=1.6]{0.01}{5}
+ \psaxes[Dy=0.1]{->}(0,0)(5,0.75)
+\end{pspicture*}
+\end{LTXexample}
+
+
+
+
+
+\clearpage
+
\subsection{Beta distribution}
A general type of statistical distribution which is related to the gamma distribution.