Bivariate normal distribution

We would like to draw a bivariate normal distribution and show where the means from the two variables meet in the space.

There are a couple of things in the code that might be useful:

You can define mathematical functions using declare function={<name>(<argument macros>)=<function>;}, which will help to keep the code clean and avoid repetitions.

You can define a new colormap using \pgfplotsset{colormap={<name>}{<color model>(<distance>)=(<value1>); <color model>(<distance 2>)=(<value2>)} }. This is a very powerful feature, so definitely worth reading in the pgfplots manual.

The legend created by colorbar is a whole new plot, so you can configure it with all the usual axis options.

There are different ways for defining 3D functions: \addplot3 {<function>}; will evaluate <function> at every point on a grid and assume the result to be a z-value. \addplot3 ({<x>},{<y>},{<z>}); defines a parametric function in 3D space, which allows you to draw three-dimensional lines, among other things.

Bivariate normal distribution

Edit and compile if you like:
  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;}]
    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 style = {
      at     = {(1,0)},
      anchor = south west,
      height = 0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
      title  = {$P(x_1,x_2)$}
    \addplot3 [surf] {bivar(mu1,sigma1,mu2,sigma2)};
    \addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth]
    \addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth]

    \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:$P(x_2)$] {};
Click to download: bivariate-normal-distribution.texbivariate-normal-distribution.pdf