Probabilités: loi normale - Fonction exponentielle, intégrales et algorithme (Monté Carlo) - Vrai/faux: géométrie dans l'espace - Suite récurrente définie avec une fonction homograhique

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Bac S 2019, métropole

Fonction exponentielle, intégrales - Probabilités: lois uniforme et normale, arbre de probabilités, suite de probabilités - Vrai/faux: complexes, asymptote, algorithme - Géométrie dans l'espace

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Bac S 2019, Amérique du Nord

Probabilités, loi normale et arbre pondéré de probabilités - Vrai / faux: nombres complexes - Fonction logarithme (ln) et suite récurrente - Géométrie dans l'espace

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    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�,
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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. $X$
\[\begin{tabular}{*6{|c}|}\hline
$x_i$ & $x_1$ & $x_2$ & $x_3$& \dots & $x_n$ \\\hline
$p_i=\text{Prob}(X=x_i)$ & $p_1$ & $p_2$ & $p_3$ & \dots & $p_n$ \\\hline
\end{tabular}\]

&Densit� de probabilit� $f$ %de la v.a. $X$, 
d�finie sur $[a;b]$


\vspace*{-1em}
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&&&\\
$f$&\psline{->}(0.2,-0.2)(1.,0.6)&\psline{->}(0.2,0.6)(1.,-0.2)&\\
&&&\\\hline
\end{tabular}\]
%\psline(-15,-0.5)(10,-0.5)
%\psline(-15,-1.5)(10,-1.5)
\\

\vspace{-.6cm}
\bgit
\item[$\bullet$] pour tout $1\leqslant i\leqslant n$,\quad $p_i\geqslant 0$ 
\vsp
\item[$\bullet$] $\dsp p_1+p_2+\cdots+p_n=\sum_{i=1}^n p_i=1$ 
\enit

&
\vspace*{-.7cm}
\bgit
\item[$\bullet$] pour tout $x\in\R$,\quad $f(x)\geqslant 0$ 
\vsp
\item[$\bullet$] $\dsp\int_{a}^{b} f(x)\,dx=1$ 
\enit
\\

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& 
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\\
\vspace*{-1em}
\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

&
\vspace*{-1.2em}\bgit
\item[$\bullet$] Esp�rance: $\dsp E(X)=\int_{a}^{b} x\,f(x)\,dx$
  \vspd
\item[$\bullet$] Variance: 
  $\bgar[t]{ll} 
  V(X)
  &\dsp=E\Bigr((X-E(X)^2\Bigl)\\
  &\dsp=\int_{a}^{b} \lp x-E(X)\rp^2\,f(x)\,dx
  \enar$
  \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}
$\bgar[t]{ll}
F(x)&=P(X\!\leqslant\! x)\\[0.2cm]
&\dsp=\int_{a}^x f(t)\,dt
\enar$
\vspd

\[\begin{tabular}{|c|ccc|}\hline
\ \ $x$\ \ & $a$ &\hspace*{1.5cm}& $b$ \\\hline
&&&\\
$f$&\psline{->}(0.2,-0.2)(1.,0.6)&\psline{->}(0.2,0.6)(1.,-0.2)&\\
&&&\\\hline
&&&$1$\\
$F$ &\psline{->}(0.2,-0.2)(2.2,0.6)&&\\
&$0$ && \\\hline
\end{tabular}
\]
\\
\vspace*{-2.5em}
\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
\\

$\dsp P(X=k)=C_n^k p^k (1-p)^{n-k}$

&
$\dsp f(x)
=\dfrac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-m)^2}{2\sigma^2}}$\ \  
sur $\R$, 
\quad
$P(X=a)=0$

\\

$\dsp 
P(X\leqslant N)
=\sum_{k=1}^N P(X=k) 
=\sum_{k=1}^N C_n^k p^k (1-p)^{n-k}$


& \ul{Loi $\mathcal{N}(0;1)$:} \ \ 
$\dsp P(X\leqslant a)=\int_{-\infty}^a f(x)\,dx=\Pi(a)$ 

\\
$\dsp 
P(N_1\leqslant X\leqslant N_2)
=\sum_{k=N_1}^{N_2} C_n^k p^k (1-p)^{n-k}$

&
\qquad $\dsp P(a<X\leqslant b)=\int_{a}^b f(x)\,dx=\Pi(b)-\Pi(a)$ 


\end{tabular}


\label{LastPage}
\end{document}

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