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\begin{document}


\begin{poster}
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%----------------------------------------------------------------------
%	EN-TÊTE
%----------------------------------------------------------------------
{\mbox{}\hspace*{0.5cm}\includegraphics[height=2cm]{Probas_continues.eps}} %gauche
{\mbox{}{\Huge Probabilités continues}}{\footnotesize Terminale S} %titre
{\includegraphics[height=2cm]{Logo_Nath.eps}\hspace*{0.5cm}\mbox{}}%droite
% 

%---------------------
% Fiche
%---------------------

\headerbox{\bf Variable aléatoire à densité sur $\bm{I}$} %%%%%%%
{name=vad,column=0,span=2}
{
Fonction de densité sur $I$ : fonction $f$ continue et positive sur $I$ telle que \\ [1mm]
\hspace*{1.75cm} $\displaystyle\int_If(t)\,\text{d}t=1$
\begin{itemize}
   \item $P(a\leqslant X \leqslant b)=\displaystyle\int_{a}^{b}f(t)\,\text{d}t$
   \item $P(X=a)=0$
   \item $P(a\leqslant X \leqslant b) \!=P(a< X \leqslant b) \!=P(a\leqslant X < b) =P(a< X < b)$
   \item $P(X\leq t) =1-P(X\geq t)$
   \item $E(X) =\displaystyle{\int_Itf(t)\,\text{d}t}$ \smallskip
\end{itemize}
}

\headerbox{\bf Loi uniforme sur \bm{$[\,a\,,b\,]$}} %%%%%%%
{name=lun,column=2,span=2,bottomaligned=vad}
{
Notation : $\mathcal U\,[\,a\,,b\,]$ \\
$f(t) =\dfrac{1}{b-a}$ \\
$\text{P}(c\leq X\leq d)=\dfrac{d-c}{b-a}$ \\
$E(X)=\dfrac{a+b}{2}$ \\
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\headerbox{\bf Loi exponentielle sur $\bm{\R^+}$} %%%%%%%
{name=lex,column=4,span=2,bottomaligned=vad}
{
Notation : $\mathcal{E}(\lambda)$ \\
$f(t)=\lambda e^{-\lambda t}$ \quad avec $\lambda>0$ \\
$P(a\leqslant X \leqslant b) =\e^{-\lambda a}-\e^{-\lambda b}$ \\
$P(X\leq t)=1-e^{-\lambda t}$ \\
$P(X\geq t)=e^{-\lambda t}$ \\
$P_{X\geq t}(X\geq t+h) =P(X\geq h)$ \\ [0.5mm]
$E(X)=\dfrac{1}{\lambda}$ \\
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\headerbox{\bf Loi normale centrée réduite sur $\bm{\R}$} %%%%%%%
{name=lnc,column=2,span=2,below=vad}
{
Notation : $\mathcal{N}(0;1)$ \\
$f(t) =\dfrac{1}{\sqrt{2\pi}}e^{-\frac12t^2}$ \\ [1mm]
$E(X) =0$ et $V(X) =1$ \\
$\forall\,\alpha\in\,]\,0\,;\,1\,[, \exists\,!\,u_\alpha\in\R_*^+$ tel que \\
$P(-u_\alpha\leqslant X\leqslant u_\alpha) =1-\alpha$ \\
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$P(-1,96\leqslant X\leqslant1,96) =0,95$ \\
$P(-2,58\leqslant X\leqslant2,58) =0,99$
}

\headerbox{\bf Loi normale sur $\bm{\R}$} %%%%%%%
{name=lno,column=4,span=2,below=vad,bottomaligned=lnc}
{
Notation : ${\cal N}(\mu;\sigma^2)$ \\
$E(X) =\mu$ et $V(X) =\sigma^2$ \\
$X$ suit la loi $\mathcal{N}(\mu;\sigma^2) \iff$ \\ [1mm]
$Z=\dfrac{X-\mu}{\sigma}$ suit la loi $\mathcal{N}(0\,;\,1)$ \\
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$P(\mu-\sigma\leq X\leq\mu+\sigma) \approx0,68$ \\
$P(\mu-2\sigma\leq X\leq\mu+2\sigma) \approx0,96$ \\
$P(\mu-3\sigma\leq X\leq\mu+3\sigma) \approx0,997$
}

\headerbox{\bf Théorème de Moivre-Laplace} %%%%%%%
{name=tml,column=0,span=2,below=vad,bottomaligned=lnc}
{
\begin{itemize}
   \item $p\in\,]\,0\,;\,1\;[$ et $n\in\N^*$
   \item $X_n$ suit la loi binomiale $\mathcal{B}(n;p)$
   \item $Z_n =\dfrac{X_n-np}{\sqrt{npq}}$ \medskip
\end{itemize}
Pour tous réels $a$ et $b$ tels que $a \leq b$ :
\begin{center}
   $\displaystyle{\lim_{n\to+\infty} P\left(a \leq Z_n \leq b\right)}$ \\
   $\displaystyle{=\int_a^b \frac{1}{\sqrt{2\pi}} e^{-\frac{t^2}{2}} \text{d}t}$
\end{center}
}

\headerbox{\bf Propriétés des lois normales} %%%%%%%
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\headerbox{\bf Intervalle de fluctuation} %%%%%%%
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Intervalle de fluctuation asymptotique au seuil de $1-\alpha$
\begin{center}
   $I_n =\left[\,p-u_\alpha\sqrt{\dfrac{p(1-p)}{n}}\,;\,p+u_\alpha\sqrt{\dfrac{p(1-p)}{n}}\,\right]$
\end{center}
Conditions : $n\geq30 \; ; \; np \geq5 \; ; \; n(1-p) \geq5$ \\
à 95\,\% : $u_{0,05} =1,96$ \quad et \quad à 99\,\% : $u_{0,01} =2,58$
}

\headerbox{\bf Intervalle de confiance} %%%%%%%
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Intervalle de confiance de $p$ au niveau de confiance 95\,\%
\begin{center}
   $\left[f-\dfrac{1}{\sqrt{n}}\,;\,f+\dfrac{1}{\sqrt{n}}\right]$
\end{center}
Conditions : $n\geq30 \; ; \; nf \geq5 \; ; \; n(1-f) \geq5$
}

\end{poster}
\end{document}


