\begin{propriete}[Quelques intervalles remarquables]
Soit $X$ une variable aléatoire suivant la loi normale $\mathscr{N}(\mu\ ;\ \sigma^2)$. On a alors  :
\begin{itemize}
	\item $P(X\in[\mu-\sigma\ ;\ \mu+\sigma])=P(\mu-\sigma\leqslant X\leqslant \mu+\sigma)\approx\nombre{0,68}$ ;
	\item $P(X\in[\mu-2\sigma\ ;\ \mu+2\sigma])=P(\mu-2\sigma\leqslant X\leqslant \mu+2\sigma)\approx\nombre{0,954}$ ;
	\item $P(X\in[\mu-3\sigma\ ;\ \mu+3\sigma])=P(\mu-3\sigma\leqslant X\leqslant \mu+3\sigma)\approx\nombre{0,997}$.
\end{itemize}

Graphiquement, on a alors :

\vspace{-1\baselineskip}\begin{center}
\begin{tikzpicture}[general,xscale=0.5,yscale=10]
\fill[color=A3](3,0)--(3,0.08066) -- plot [domain=3:9] (\x,{0.133*exp{-0.0556*(\x-6)*(\x-6)}})--(9,0.08066)-- (9,0) -- cycle;

\fill[color=H1](9,0) -- (9,0.08066)-- plot [domain=9:12] (\x,{0.133*exp{-0.0556*(\x-6)*(\x-6)}})-- (12,0.018)--(12,0) -- cycle;
\fill[color=H1](3,0) -- (3,0.08066)-- plot [domain=3:0] (\x,{0.133*exp{-0.0556*(\x-6)*(\x-6)}})-- (0,0.018)--(0,0) -- cycle;

\fill[color=J1](12,0) -- (12,0.018)-- plot [domain=12:15] (\x,{0.133*exp{-0.0556*(\x-6)*(\x-6)}})-- (15,0.00148)--(15,0) -- cycle;
\fill[color=J1](0,0) -- (0,0.018)-- plot [domain=0:-3] (\x,{0.133*exp{-0.0556*(\x-6)*(\x-6)}})-- (-3,0.00148)--(-3,0) -- cycle;

\draw[dotted] (6,0)--(6,0.133);


\axeX{-6}{18}{};
%\axeY{0}{0.25}{};
\draw[domain=-6:18,epais,samples=100] plot(\x,{0.133*exp{-0.0556*(\x-6)*(\x-6)}});
\draw(6,0) node[below] {{\small $\mu$}};
\draw(9,0) node[below] {{\small $\mu+\sigma$}};
\draw(12,0) node[below] {{\small $\mu+2\sigma$}};
\draw(15,0) node[below] {{\small $\mu+3\sigma$}};
\draw(3,0) node[below] {{\small $\mu-\sigma$}};
\draw(0,0) node[below] {{\small $\mu-2\sigma$}};
\draw(-3,0) node[below] {{\small $\mu-3\sigma$}};


\draw(9,0) node {{\small $+$}};
\draw(12,0) node {{\small $+$}};
\draw(15,0) node {{\small $+$}};
\draw(3,0) node {{\small $+$}};
\draw(0,0) node {{\small $+$}};
\draw(-3,0) node {{\small $+$}};
\end{tikzpicture}
\end{center}

\vspace{-\baselineskip}où l'aire du domaine en bleu est environ \nombre{0,68}, l'aire du domaine en bleu et vert est environ \nombre{0,954} et l'aire du domaine en bleu, vert et orange (jusqu'à $\mu-3\sigma$ et $\mu+3\sigma$) est environ \nombre{0,997}.
\end{propriete}

