Could TiKz make a picture of a matrix with pictorial normal distributions?

If you do not want that much control and precision over and in your little pictures, you can use the following solution.

It uses

• amstext (usually loaded by amsmath anyway) for the \text macro,
• blkarray for the parentheses between columns (there could easily by various TikZ(mark) solutions usable here), and
• PGF/TikZ

The TikZ picture consists of two paths for the axes and one path for the plot.

The picture’s usual coordinate system is shrinked by the factor 10. I took a rather unorthodox way for the axes and just used coordinates of the factor 1 (base unit). The actual x and the y value are computated via the x and y factor.

This way, the axes do not scale with the plot or the total TikZ picture but after all the domain setting also affect the plot. A special style my be introduced here that sets all other settings based on the domain. I have used two examples with borderline values so that either the y axis had to be extended or the whole picture has to be shrunken which probably won’t be a real use-case as this distorts any relation to the other pictures.

Be aware of a high samples value because this slows down the process very much. It may be advisavle to externalize the calculation (gnuplot) and/or use the external library of TikZ. You also may pre-calculate some values (you’ll need a table plot then) and simply change the coordinate system and/or scale the resulting picture.

Code

\documentclass{article}
\usepackage{amstext}
\usepackage{blkarray}
\usepackage{tikz}
\tikzset{
  declare function={NormDist(\xValue,\meanValue,\standardDeviation)=1/(\standardDeviation*sqrt(2*pi))*exp(-((\xValue-\meanValue)^2)/(2*\standardDeviation^2));},
}
\newcommand*{\tikzNormDist}[3][]{%
  \begin{tikzpicture}[every mini plot picture/.try,#1]
    \path[every mini plot x axis/.try] (-1,0) -- (1,0);
    \path[every mini plot y axis/.try] (0,0) -- (0,1) [mini plot y axis arrow/.try];
    \draw[every mini plot plot/.try]   plot[variable=\x] (\x,{NormDist(\x,#2,#3)});
  \end{tikzpicture}
}
\tikzset{
  every mini plot picture/.style={x=+1mm, y=+1mm, baseline=.5},
  every mini plot x axis/.style={draw,thin,x=+2mm},
  every mini plot y axis/.style={draw,thin,y=+2.2mm},
  every mini plot plot/.style={
    thick,
    red,
    smooth,
    samples=40,
    domain=-2:2,
  },
  mini plot y axis arrow/.style={
    insert path={
      [line cap=round, line join=round]+(+.25mm,+-.5mm) -- +(+0,+0) -- +(+-.25mm,+-.5mm)
    }
  }
}
\begin{document}
\[
  \begin{blockarray}{l c*2{@{\hphantom{{}+{}}}c}}
   & \text{Table} & \text{Bed} & \text{Pat} \\
  \begin{block}{l(*2{c@{{}+{}}}c)}
  \text{Table} & \tikzNormDist{0}{0.3}   & \tikzNormDist[
                                               every mini plot plot/.append style={blue!50!red,fill=blue!50!green},
                                               every mini plot y axis/.append style={y=+3.5mm}
                                           ]{-.3}{0.15}               & \tikzNormDist{-.2}{0.4} \\
  \text{Bed}   & \tikzNormDist{.2}{0.5}  & \tikzNormDist{.7} {0.7}    & \tikzNormDist{-.5}{0.7} \\
  \text{Pat}   & \tikzNormDist{-.1}{0.4} & \tikzNormDist{-.1}{1}      & \tikzNormDist[
                                                                            every mini plot plot/.append style={domain=-10:10,x=+.25mm,y=+5mm}
                                                                        ]{0}{2} \\
  \end{block}
  & {} = 1 & {} = 1 & {} = 1 \\
  \end{blockarray}
\]
\end{document}

Output