Since PGFPlots version 1.10 came out, there’s a more elegant way to do this, anyway this is good for learning.
\documentclass[tikz,border=10pt]{standalone}
\usepackage{tkz-fct}
\usetikzlibrary{intersections}
\begin{document}
\tikzset{
name plot/.style={every path/.style={name path global=#1}}
}
% Extract coordinates for point X
\makeatletter
\newcommand{\gettikzxy}[3]{%
\tikz@scan@one@point\pgfutil@firstofone#1\relax
\edef#2{\the\pgf@x}%
\edef#3{\the\pgf@y}%
}
\makeatother
% Dimlines
\def\Dimline[#1][#2][#3][#4]{
\begin{scope}[thin, >=stealth'] % redefine as flechas
\draw let \p1=#1, \p2=#2, \n0={veclen(\x2-\x1,\y2-\y1)} in [|<->|,
decoration={markings,mark=at position .5 with {\node[#3] at (0,0)
{#4};},
},
postaction=decorate] #1 -- #2 ;
\end{scope}
}
\begin{tikzpicture}[scale=1,font=\small]
\tkzInit[xmin=0,xmax=12,ymin=0,ymax=.3,ystep=.05]
% Draw coordinates
\draw[>=stealth', <->] (0,6) node[above] {$y$} -- (0,0) -- (12.5,0) node[right] {$x$};
% Draw functions and areas
\tkzFct[name plot=A,thick,color=red,domain=0:12]{1/(1.5*sqrt(2*pi))*exp(-((x-4.5)**2)/(2*1.5**1))}
\tkzDrawArea[opacity=.3,color=blue,domain = 7:12]
\tkzFct[name plot=B,thick,color=blue,domain=0:12]{1/(2*sqrt(2*pi))*exp(-((x-7)**2)/(2*2**1))}
\tkzDrawArea[opacity=.3,color=red,domain=0:7]
\tkzDrawAreafg[between=b and a,opacity=.3,color=green,domain = 0:7]
% Intersection between curves
\path [name intersections={of=A and B,by=C}];
% Extract coordinates of C
\gettikzxy{(C)}{\cx}{\cy}
% Vertical lines
\draw [thick,dashed, black] (\cx,0) -- (\cx,5.5) node [above] {$x_{0}$};
\draw [thick,dashed, black] (7,0) -- (7,5.5) node [above] {$\hat{x}$};
% Define regions
\Dimline[($(0,0)+(0,-.6)$)][($(7,0)+(0,-.6)$)][above,black][$\mathcal{R}_{1}$];
\Dimline[($(7,0)+(0,-.6)$)][($(12,0)+(0,-.6)$)][above, black][$\mathcal{R}_{2}$];
\end{tikzpicture}
\end{document}