How to draw a sample from a probability function and the show it in a tikz surf plot?

I saw this post bivariate which plots a bivariate normal probability density function. What I want is to draw (statistically) a number of points based on this probability function and then show it in the same plot as the beautiful plot in the aforementioned question.

The following is a the source code from the above source:

\documentclass{standalone}

\usepackage{pgfplots}

\begin{document}

\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)}
}

\begin{tikzpicture}[
declare function={mu1=1;},
declare function={mu2=2;},
declare function={sigma1=0.5;},
declare function={sigma2=1;},
declare function={normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
declare function={bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;}]
\begin{axis}[
colormap name=whitered,
width=15cm,
view={45}{65},
enlargelimits=false,
grid=major,
domain=-1:4,
y domain=-1:4,
samples=26,
xlabel=$x_1$,
ylabel=$x_2$,
zlabel={$P$},
colorbar,
colorbar style={
at={(1,0)},
anchor=south west,
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
title={$P(x_1,x_2)$}
}
]
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (x,4,{normal(mu1,sigma1)});
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (-1,x,{normal(mu2,sigma2)});

\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);

\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}


What I want is to draw a number of points (say 100) from the probability distribution, and then draw these points on the same or another plot.

• Could you please provide an MWE?
– user156344
Jan 6, 2019 at 11:13
• I changed the link to the original source which draws a bivariate normal. Jan 6, 2019 at 11:14
• @JouleV, what is an MWE? Jan 6, 2019 at 11:16
• Please help us help you and add a minimal working example (MWE) that illustrates your problem. Reproducing the problem and finding out what the issue is will be much easier when we see compilable code, starting with \documentclass{...} and ending with \end{document}.
– DG'
Jan 6, 2019 at 12:11
• @DG', thank you very much. I hope this makes the problem more clear. Jan 6, 2019 at 13:11

This is one option, I just used this other post to generate the numbers

\documentclass[border=10pt]{standalone}

\usepackage{pgfplots}

% https://tex.stackexchange.com/questions/148091/gaussian-random-numbers
\newcount\gaussF
\edef\gaussR{0}
\edef\gaussA{0}

\makeatletter
\pgfmathdeclarefunction{invgauss}{2}{%
\pgfmathln{#1}% <- might need parsing
\pgfmathmultiply@{\pgfmathresult}{-2}%
\pgfmathsqrt@{\pgfmathresult}%
\pgfmathmultiply{6.28318531}{#2}% <- might need parsing
\pgfmathdeg@{\pgfmathresult}%
\pgfmathcos@{\pgfmathresult}%
}

\pgfmathdeclarefunction{randnormal}{0}{%
\pgfmathrnd@
\ifdim\pgfmathresult pt=0.0pt\relax%
\def\pgfmathresult{0.00001}%
\fi%
\let\@tmp=\pgfmathresult%
\pgfmathrnd@%
\ifdim\pgfmathresult pt=0.0pt\relax%
\def\pgfmathresult{0.00001}%
\fi
\pgfmathinvgauss@{\pgfmathresult}{\@tmp}%
}

\makeatother

\pgfplotsset{%
colormap={whitered}{color(0cm)=(white);
color(1cm)=(orange!75!red)}
}
\begin{document}
\begin{tikzpicture}[
declare function = {mu1=1;},
declare function = {mu2=2;},
declare function = {sigma1=0.5;},
declare function = {sigma2=1;},
declare function = {normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
declare function = {bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;}]
\begin{axis}[
colormap name  = whitered,
width          = 15cm,
view           = {45}{65},
enlargelimits  = false,
grid           = major,
domain         = -1:4,
y domain       = -1:4,
samples        = 26,
xlabel         = $x_1$,
ylabel         = $x_2$,
zlabel         = {$P$},
xmin           = -1,
xmax           = 4,
ymin           = -1,
ymax           = 4,
colorbar,
colorbar style = {
at     = {(1,0)},
anchor = south west,
height = 0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
title  = {$P(x_1,x_2)$}
}
]
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth]
(x,4,{normal(mu1,sigma1)});
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth]
(-1,x,{normal(mu2,sigma2)});

\pgfplotsinvokeforeach{1,...,500} {
\pgfmathsetmacro{\xx}{mu1 + sigma1 * randnormal}
\pgfmathsetmacro{\yy}{mu2 + sigma2 * randnormal}
\addplot3 [mark=*, mark size=1pt] coordinates {(\xx, \yy, 0)};
}

\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);

\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}

\end{tikzpicture}
\end{document}


Another proposal based on the inverse Gaussian trick.

\documentclass{standalone}

\usepackage{pgfplots}

\begin{document}

\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)}
}

\begin{tikzpicture}[
declare function={mu1=1;
mu2=2;
sigma1=0.5;
sigma2=1;
normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));
bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;
mybivar(\x,\y,\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((\x-\ma)^2/\sa^2 + (\y-\mb)^2/\sb^2))/2;
invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b));}]
% invgauss based on https://tex.stackexchange.com/questions/254484/how-to-make-a-graph-of-heteroskedasticity-with-tikz-pgf/254494#254494
\foreach\X in {1,...,100}
{\pgfmathsetmacro{\myx}{mu1+0.5*sigma1*invgauss(rnd,rnd)}
\pgfmathsetmacro{\myy}{mu2+0.5*sigma2*invgauss(rnd,rnd)}
\pgfmathsetmacro{\myz}{mybivar(\myx,\myy,mu1,sigma1,mu2,sigma2)}
\ifnum\X=1
\xdef\lstOne{(\myx,\myy,\myz)}
\else
\xdef\lstOne{\lstOne (\myx,\myy,\myz)}
\fi}
\begin{axis}[
colormap name=whitered,
width=15cm,
view={45}{65},
enlargelimits=false,
grid=major,
domain=-1:4,
y domain=-1:4,
samples=26,
xlabel=$x_1$,
ylabel=$x_2$,
zlabel={$P$},
colorbar,
colorbar style={
at={(1,0)},
anchor=south west,
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
title={$P(x_1,x_2)$}
}
]
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (x,4,{normal(mu1,sigma1)});
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (-1,x,{normal(mu2,sigma2)});

\addplot3 [only marks, mark=*, mark layer=like plot,mark size=1pt] coordinates {\lstOne};

\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);

\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}