Source Latex
du cours de mathématiques
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\hypersetup{
pdfauthor={Yoann Morel},
pdfsubject={Synth�se probabilit�s: variables al�atoires discr�tes
et continues},
pdftitle={Variables al�atoires discr�tes et continues},
pdfkeywords={Probabilit�s, Variables al�atoires discr�tes,
Variables al�atoires continues, v.a.,
discret, continu, discr�tes, continues,
loi de probabilit�, loi continue, densit� de probabilit�,
fonction de r�partition, loi binomiale, loi normale, gaussienne,
gauss, loi normale centr�e r�duite, loi exponentielle}
}
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\author{Y. Morel}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
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\hspace{-0.5cm}%\noindent
\begin{tabular}{p{0.52\linewidth}|p{0.5\linewidth}}
Loi de probabilit� $P$ de la v.a. $X$
\[\begin{tabular}{*6{|c}|}\hline
$x_i$ & $x_1$ & $x_2$ & $x_3$& \dots & $x_n$ \\\hline
$\text{Prob}(X=x_i)$ & $p_1$ & $p_2$ & $p_3$ & \dots & $p_n$ \\\hline
\end{tabular}\]
&Densit� de probabilit� $f$ de la v.a. $X$ (d�finie sur $\R$)
\[\begin{tabular}{|c|ccc|}\hline
$x$ & $-\infty$ &\hspace*{1cm}& $+\infty$ \\\hline
&&&\\
$f(x)$&\psline{->}(0.2,-0.2)(1.,0.6)&\psline{->}(0.2,0.6)(1.,-0.2)&\\
&0&&0\\\hline
\end{tabular}\]
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\\
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\bgit
\item[$\bullet$] pour tout $1\leqslant i\leqslant n$,\quad $p_i\geqslant 0$
\vsp
\item[$\bullet$] $\dsp\sum_{i=1}^n p_i=1$
\enit
&
\vspace{-0.4cm}
\bgit
\item[$\bullet$] pour tout $x\in\R$,\quad $f(x)\geqslant 0$
\vsp
\item[$\bullet$] $\dsp\int_{-\infty}^{+\infty} f(x)\,dx=1$
\enit
\\
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&
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\\
\bgit
\item[$\bullet$] Esp�rance: $\dsp E(X)=\sum_{i=1}^n x_i\,p_i$
\vspd
\item[$\bullet$] Variance:
$\bgar[t]{ll}
V(X)
&\dsp=E\Bigr((X-E(X)^2\Bigl)\\
&\dsp=\sum_{k=1}^n \lp x-E(X)\rp^2\,p_i
\enar$
\vspd
\item[$\bullet$] Ecart type: $\sigma=\sqrt{V(X)}$
\enit
&
\bgit
\item[$\bullet$] Esp�rance: $\dsp E(X)=\int_{-\infty}^{+\infty} x\,f(x)\,dx$
\vspd
\item[$\bullet$] Variance:
$\bgar[t]{ll}
V(X)
&\dsp=E\Bigr((X-E(X)^2\Bigl)\\
&\dsp=\int_{-\infty}^{+\infty} \lp x-E(X)\rp^2\,f(x)\,dx
\enar$
\vspd
\item[$\bullet$] Ecart type: $\sigma=\sqrt{V(X)}$
\enit
\\
\noindent
\ul{Probabilit�s cumul�es croissantes}
\[P(X\!\leqslant\! x_i)=\sum_{k=1}^i p_k\]
\[\begin{tabular}{*7{|c}|}\hline
\multicolumn{2}{|c|}{$x_i$} & $x_1$ & $x_2$ & $x_3$& \dots & $x_n$ \\\hline
\multicolumn{2}{|c|}{$P(X=x_i)$} & $p_1$ & $p_2$ & $p_3$ & \dots &$p_n$
\\\hline
$P(X\leqslant x_i)$&$0$ & $p_1$ & $p_1\!+\!p_2$ &
$p_1\!+\!p_2\!+\!p_3$ & $\dots$ & $1$\\\hline
\end{tabular}
\]
&
\ul{Fonction de r�partition}
$\bgar[t]{ll}
F(x)&=P(X\!\leqslant\! x)\\[0.2cm]
&\dsp=\int_{-\infty}^x f(t)\,dt
\enar$
\vspd
\[\begin{tabular}{|c|ccc|}\hline
$x$ & $-\infty$ &\hspace*{1cm}& $+\infty$ \\\hline
&&&\\
$f(x)$&\psline{->}(0.2,-0.2)(1.,0.6)&\psline{->}(0.2,0.6)(1.,-0.2)&\\
&0&&0\\\hline
&&&$1$\\
$F(x)$ &\psline{->}(0.2,-0.2)(2.2,0.6)&&\\
&$0$ && \\\hline
\end{tabular}
\]
\\
\ul{Loi binomiale $\mathcal{B}(n,p)$} \quad
$E(X)=np$, $\sigma(X)=\sqrt{npq}$
%\vsp
&\ul{Loi normale $\mathcal{N}(m,\sigma)$}\quad
$E(X)=m$, $\sigma(X)=\sigma$
%\vsp
\\
$\dsp P(X=k)=C_n^k p^k (1-p)^{n-k}$
&
$\dsp f(x)
=\dfrac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-m)^2}{2\sigma^2}}$, \quad
$P(X=a)=0$
\\
$\dsp
P(X\leqslant N)
=\sum_{k=1}^N P(X=k)
=\sum_{k=1}^N C_n^k p^k (1-p)^{n-k}$
&$\dsp P(X\leqslant a)=\int_{-\infty}^a f(x)\,dx=\Pi(a)$
\\
$\dsp
P(N_1\leqslant X\leqslant N_2)
=\sum_{k=N_1}^{N_2} C_n^k p^k (1-p)^{n-k}$
&
$\dsp P(a<X\leqslant b)=\int_{a}^b f(x)\,dx=\Pi(b)-\Pi(a)$
\\
\end{tabular}
\label{LastPage}
\end{document}
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