\documentclass[varwidth=true, border=5pt]{article}
\usepackage[active,tightpage]{preview}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{arrows,positioning}
\usepackage{helvet}
\usepackage[eulergreek]{sansmath}

\pgfmathdeclarefunction{gauss}{2}{%
  \pgfmathparse{1/(sqrt(2*pi*#2))*exp(-((x-#1)^2)/(2*#2))}%
}
% to be used: \gauss(x)(mean)(variance)

\makeatletter \newcommand{\pgfplotsdrawaxis}{\pgfplots@draw@axis} \makeatother

\pgfplotsset{
tick label style = {font=\sansmath\sffamily},
every axis label/.append style={font=\sffamily\footnotesize},
}

\pgfplotsset{axis line on top/.style={
  axis line style=transparent,
  ticklabel style=transparent,
  tick style=transparent,
  axis on top=false,
  after end axis/.append code={
    \pgfplotsset{axis line style=opaque,
      ticklabel style=opaque,
      tick style=opaque,
      grid=none}
    \pgfplotsdrawaxis}
  }
}

\tikzset{>=latex}

\newcommand{\meanI}{80}
\newcommand{\meanII}{40}
\newcommand{\varI}{40}
\newcommand{\varII}{40}

\newcommand{\arrowHeight}{0.02}
\newcommand{\labelheight}{0.055}

\pgfmathsetmacro{\meanIheight}{1/(sqrt(2*pi*\varI))}%
\pgfmathsetmacro{\meanIIheight}{1/(sqrt(2*pi*\varII))}%
\pgfmathsetmacro{\plotheight}{max(\meanIheight, \meanIIheight, \labelheight+0.002)}%

\begin{document}
\begin{preview}
\tikzstyle{plotA}=[ultra thick,red!90!black]
\tikzstyle{plotB}=[ultra thick,cyan!50!black]
\begin{tikzpicture}
    \begin{axis}[
        width=13.5cm,
        height=8.625cm,
        % Grid
        grid = major,
        grid style={thin, dashed},
        % size
        xmin= 20,     % start the diagram at this x-coordinate
        xmax= 105,   % end   the diagram at this x-coordinate
        ymin= 0,     % start the diagram at this y-coordinate
        ymax= \plotheight, % end   the diagram at this y-coordinate
        % Legende
        legend style={
            font=\large\sansmath\sffamily,
            at={(0.5,-0.18)},
            anchor=north,
            legend cell align=left,
            legend columns=-1,
            column sep=0.5cm
        },
        % Ticks
        tick align=inside,
        scaled y ticks = false,
        ytick=\empty,
        % Axis
        axis lines = middle,
        axis line style = very thick,
        axis line on top,
        axis line style = {very thick,shorten <=-0.5\pgflinewidth},
        xlabel=$x$,
        x label style={at={(axis description cs:0.52,0)},
                       anchor=north,
                       font=\boldmath\Large},
        ylabel=$f(x)$,
        y label style={at={(axis description cs:0,0.5)},
                       anchor=south,
                       rotate=90,
                       font=\boldmath\Large},
        ]

        % Mark means
        \coordinate (axisOrigin) at (axis cs:0,0);
        \coordinate (meanA) at (axis cs:\meanI,\meanIheight);
        \draw[black,thick](meanA |- axisOrigin) -- (meanA);

        \coordinate (meanB) at (axis cs:\meanII,\meanIIheight);
        \draw[black,thick](meanB |- axisOrigin) -- (meanB);
        \draw[black,ultra thick,<->](axis cs:{\meanI-sqrt(\varI)}, \arrowHeight) -- (axis cs:{\meanI+sqrt(\varI)},\arrowHeight);

        % Plots of the distributions
        \addplot [domain=0:120,samples=400,plotA] {gauss(\meanI,\varI)};
        \addplot [domain=0:120,samples=400,plotB] {gauss(\meanII,\varII)};

        % Add labels
       \node[plotA,above] at (axis cs:\meanI+11,\labelheight){\Large $\mathcal{N}(\meanI, \varI)$};
       \node[plotB,above] at (axis cs:\meanII-11,\labelheight){\Large $\mathcal{N}(\meanII, \varII)$};
    \end{axis}
\end{tikzpicture}
\end{preview}
\end{document}