# Student t-distribution with TikZ

I'd like to draw a Student's t-distribution with five degrees of freedom using TikZ, then another with 10 degrees of freedom, etc.

In the program I am working in the degrees of freedom will be a random number from 1 to 20, so I need a t-distribution for the degrees of freedom assigned by Perl randomization.

## 5 Answers

This is a pgfplots/gnuplot solution.

For \addplot gnuplot {…} to work you need to have a working installation of gnuplot on your machine and have to call pdflatex with write18 enabled (i.e. --shell-escape or --enable-write18).
How pgfplots and gnuplot interact can be studied in the pgfplots manual in subsection 4.2.5 “Computing Coordinates With Mathematical Expressions (gnuplot)”.

There are two foreach loops in this code. One that loops over tikzpicture and gives you one plot per picture, the other one loops over \addplot so that you will get one picture with nineteen plots.

Edit: Apparently gnuplot sees /2 as an integer rather than a floating point division.
The function is therefore:

gamma((\n+1)/2.)/(sqrt(\n*pi)*gamma(\n/2.))*((1+(x*x)/\n)^(-(\n+1)/2.))% or
gamma((\n+1)/2.)/(sqrt(\n*pi)*gamma(\n/2.))/((1+(x*x)/\n)^((\n+1)/2.))% or
gamma(.5*(\n+1))/(sqrt(\n*pi)*gamma(.5*\n))*((1+(x*x)/\n)^(-.5*(\n+1)))% or
gamma(.5*(\n+1))/(sqrt(\n*pi)*gamma(.5*\n))/((1+(x*x)/\n)^(.5*(\n+1)))%


## Code

\documentclass[tikz,border=2pt]{standalone}
\usepackage{pgfplots}
\def\basefunc{%
gamma(.5*(\n+1))/(sqrt(\n*pi)*gamma(.5*\n))*((1+x^2/\n)^(-.5*(\n+1)))%
}
\begin{document}
\foreach \n in {1,...,20}{
\begin{tikzpicture}
\begin{axis}[
ymin=0,
ymax=.41,
]
\addplot gnuplot [
smooth,
no marks,
domain={-6:+6},
]{\basefunc};
\legend{$n = \n$}
\end{axis}
\end{tikzpicture}
}
\begin{tikzpicture}
\begin{axis}[
ymin=0,
ymax=.41,
]
\foreach \n in {2,...,20}{
\addplot gnuplot [
very thin,
smooth,
no marks,
domain={-6:+6},
]{\basefunc};
}
\end{axis}
\end{tikzpicture}
\end{document}


## Output of the second tikzpicture

If you want to plot the function without using gnuplot, you can define the approximation for the gamma function using declare function and use that for plotting the t-distribution.

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\begin{document}

\begin{tikzpicture}[
declare function={gamma(\z)=
2.506628274631*sqrt(1/\z)+ 0.20888568*(1/\z)^(1.5)+ 0.00870357*(1/\z)^(2.5)- (174.2106599*(1/\z)^(3.5))/25920- (715.6423511*(1/\z)^(4.5))/1244160)*exp((-ln(1/\z)-1)*\z;},
declare function={student(\x,\n)= gamma((\n+1)/2.)/(sqrt(\n*pi) *gamma(\n/2.)) *((1+(\x*\x)/\n)^(-(\n+1)/2.));}
]

\begin{axis}[
axis lines=left,
enlargelimits=upper,
samples=50
]
\pgfplotsinvokeforeach{1,2,5,100}{
\addplot [thick, smooth, domain=-6:6] {student(x,#1)} node [pos=0.5, anchor=mid west, xshift=2em, append after command={(\tikzlastnode.west) edge [thin, gray] +(-2em,0)}] {$n=#1$};
}
\end{axis}
\end{tikzpicture}
\end{document}


Run it with xelatex or the sequence latex->dvips->ps2pdf:

\documentclass{article}
\usepackage{pst-func}
\begin{document}

\psset{xunit=1.25cm,yunit=10cm}
\begin{pspicture*}(-6,-0.1)(6,0.5)
\psaxes[Dy=0.1]{->}(0,0)(-5,0)(5.5,0.5)
\psTDist[linewidth=1pt,plotpoints=100,linecolor=red,
fillstyle=solid,fillcolor=red!50,opacity=0.4, nue=3]{-5}{5}
\rput(3,0.3){$\nu=3$}
\end{pspicture*}

\end{document}


And for the values nue = 1,2,5,1000:

\documentclass{article}
\usepackage{pst-func}
\begin{document}

\psset{xunit=1cm,yunit=10cm}
\begin{pspicture*}(-6,-0.1)(6,0.5)
\psaxes[Dy=0.1]{->}(0,0)(-5,0)(5.5,0.5)
\psset{linewidth=1pt,plotpoints=100}
\psTDist[linecolor=red,nue=1]{-5}{5}
\psTDist[linecolor=green,nue=2]{-5}{5}
\psTDist[linecolor=blue,nue=5]{-5}{5}
\psTDist[linestyle=dashed,nue=1000]{-5}{5}
\end{pspicture*}

\end{document}


This is also possible with tkz-fct and gnuplot. It's possible to use a scope to keep the color red local for the curve. Without the scope I need to to indicate the black color to draw the axes. I used the function given by Qrrbrbirlbel. I fixed \n but it's possible to use \foreach.

Update

\documentclass[11pt]{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage{tkz-fct}
\begin{document}

\begin{tikzpicture}[scale=.6,font=\small]
\tkzInit[xmin=-5,xmax=5,ymin=-0,ymax=.5,ystep=.05]
\tkzGrid
\foreach \n in {2,4,...,20} {%
\pgfmathsetmacro\col{3*\n+20}
\tkzFct[color=red!\col!blue,domain=-5:5]{%
gamma((\n+1)/2.)/(sqrt(\n*pi)*gamma(\n/2.))*(1+x**2./\n)**(-(\n+1)/2.)}
}
\def\n{10}
\pgfmathsetmacro\col{3*\n+20}
\tkzFct[color=red!\col!blue,domain=-5:5]{%
gamma((\n+1)/2.)/(sqrt(\n*pi)*gamma(\n/2.))*(1+x**2./\n)**(-(\n+1)/2.)}
\tkzDrawArea[opacity=.3,color=red!30,domain = -6:6]
\tkzAxeXY[color=black]
\end{tikzpicture}

\end{document}


• As edited to my answer and commented by the OP gnuplot needs /2. instead of /2. – Qrrbrbirlbel Nov 4 '12 at 18:57
• @Qrrbrbirlbel Yes you are right. I used your function without checking its validity. Sorry ! – Alain Matthes Nov 6 '12 at 8:00

To complete the overkill of the number of solution here a version with gnuplottex.

\documentclass{standalone}
\usepackage{gnuplottex}
\begin{document}

\begin{gnuplot}[terminal=epslatex,terminaloptions=color solid]
unset key
set samples 1000
set format '$%g$'
set xrange [-6:6]
set yrange [0:0.41]
f(n,x) = gamma(.5*(n+1))/(sqrt(n*pi)*gamma(.5*n))*((1+x**2/n)**(-.5*(n+1)))
plot for[i=1:20] f(i,x)
\end{gnuplot}

\end{document}