@ccueil Colles

Source LaTeX icone Cours-Variables-Aleatoires-Discret-vs-Continues



Fichier
Type: Cours
File type: Latex, tex (source)
Télécharger le document pdf compilé pdficon
Description
Comparatif variables aléatoires et lois de probabilités discrètes et continues
Niveau
Terminale S
Mots clé
Cours de mathématiques, variable aléatoire, loi continue, loi discrète, probabilités, TS, terminale S
Voir aussi:

Documentation sur LaTeX
pdficon
Source LaTex icone
Télécharger le fichier source pdficon

\documentclass[12pt]{article}
%\usepackage{french}
\usepackage{amsfonts}\usepackage{amssymb}

\usepackage[french]{babel}
\usepackage{amsmath}
\usepackage[latin1]{inputenc}
\usepackage{a4wide}
\usepackage{graphicx}
\usepackage{epsf}
\usepackage{calc}
\usepackage{enumerate}
\usepackage{array}
%\usepackage{pst-plot,pst-text,pst-tree}
%\usepackage{pst-all}
\usepackage{pst-func}
\usepackage{pstricks-add}
\usepackage{colortbl}

\usepackage{hyperref}
\hypersetup{
    pdfauthor={Yoann Morel},
    pdfsubject={Synthse probabilits: variables alatoires discrtes
      et continues},
    pdftitle={Variables alatoires discrtes et continues},
    pdfkeywords={Probabilits, Variables alatoires discrtes, 
      Variables alatoires continues, v.a., 
      discret, continu, discrtes, continues, 
      loi de probabilit, loi continue, densit de probabilit,
      fonction de rpartition, loi binomiale, loi normale, gaussienne,
      gauss, loi normale centre rduite, loi exponentielle} 
}
\hypersetup{
    colorlinks = true,
    linkcolor = red,
    anchorcolor = red,
    citecolor = blue,
    filecolor = red,
    pagecolor = red,
    urlcolor = red
}
\voffset=-2.2cm
% Raccourcis diverses:
\newcommand{\nwc}{\newcommand}
\nwc{\dsp}{\displaystyle}
\nwc{\ct}{\centerline}
\nwc{\bge}{\begin{equation}}\nwc{\ene}{\end{equation}}
\nwc{\bgar}{\begin{array}}\nwc{\enar}{\end{array}}
\nwc{\bgit}{\begin{itemize}}\nwc{\enit}{\end{itemize}}
\nwc{\bgen}{\begin{enumerate}}\nwc{\enen}{\end{enumerate}}

\nwc{\la}{\left\{}\nwc{\ra}{\right\}}
\nwc{\lp}{\left(}\nwc{\rp}{\right)}
\nwc{\lb}{\left[}\nwc{\rb}{\right]}

\nwc{\bgsk}{\bigskip}
\nwc{\vsp}{\vspace{0.1cm}}
\nwc{\vspd}{\vspace{0.2cm}}
\nwc{\vspt}{\vspace{0.3cm}}
\nwc{\vspq}{\vspace{0.4cm}}

\def\N{{\rm I\kern-.1567em N}}                              % Doppel-N
\def\D{{\rm I\kern-.1567em D}}                              % Doppel-N
\def\No{\N_0}                                               % Doppel-N unten 0
\def\R{{\rm I\kern-.1567em R}}                              % Doppel R
\def\C{{\rm C\kern-4.7pt                                    % Doppel C
\vrule height 7.7pt width 0.4pt depth -0.5pt \phantom {.}}}
\def\Q{\mathbb{Q}}
\def\Z{{\sf Z\kern-4.5pt Z}}                                % Doppel Z
\def\euro{\mbox{\raisebox{.25ex}{{\it =}}\hspace{-.5em}{\sf C}}}

\renewcommand{\Re}{\mathcal{R}e}
\renewcommand{\Im}{\mathcal{I}\!m}

\def\epsi{\varepsilon}
\def\lbd{\lambda}
\def\tht{\theta}

\def\Cf{\mathcal{C}_f}

\nwc{\tm}{\times}
\nwc{\V}[1]{\overrightarrow{#1}}

\nwc{\zb}{\mbox{$0\hspace{-0.67em}\mid$}}
\nwc{\db}{\mbox{$\hspace{0.1em}|\hspace{-0.67em}\mid$}}

\nwc{\ul}[1]{\underline{#1}}

\newcounter{nex}%[section]
\setcounter{nex}{0}
\newenvironment{EX}{%
\stepcounter{nex}
\bgsk{\noindent {\bf Exercice }\arabic{nex}}\hspace{0.2cm}
}{}

\nwc{\bgex}{\begin{EX}}\nwc{\enex}{\end{EX}}

\nwc{\bgfg}{\begin{figure}}\nwc{\enfg}{\end{figure}}
  \nwc{\epsx}{\epsfxsize}\nwc{\epsy}{\epsfysize}
\nwc{\bgmp}{\begin{minipage}}\nwc{\enmp}{\end{minipage}}


\nwc{\limcdt}[4]{
  $\dsp
  \lim_{\bgar{ll}\scriptstyle{#1}\vspace{-0.2cm}\\\scriptstyle{#2}\enar}
  {#3}={#4}$
}
\nwc{\tq}{\ \mbox{\bf\Large /}\ }



\headheight=0.cm
\textheight=27.5cm
\topmargin=-1.8cm
\footskip=0.6cm
\textwidth=18.5cm
\oddsidemargin=-1.5cm
\parindent=0.2cm

\newlength{\ProgIndent}
\setlength{\ProgIndent}{0.3cm}

\setlength{\unitlength}{1cm}

\newcounter{ntheo}
\setcounter{ntheo}{1}
\newlength{\ltheo}
\nwc{\bgth}[1]{
  \settowidth{\ltheo}{Thorme \arabic{ntheo}}
  \noindent
  \paragraph{Thorme}% \arabic{ntheo}}
  \hspace{-0.5em}%\hspace{-0.4cm}
  \bgmp[t]{\textwidth-\ltheo-0.5em}{\it #1}\enmp
  \stepcounter{ntheo}
}

\newcounter{nprop}
\setcounter{nprop}{1}
\newlength{\lprop}
\nwc{\bgprop}[1]{
  \settowidth{\lprop}{Proprit \arabic{nprop}}
  \noindent
  \paragraph{Proprit}% \arabic{ntheo}}
  \hspace{-0.5em}%\hspace{-0.4cm}
  \bgmp[t]{\textwidth-\lprop-0.5em}{\it #1}\enmp
  \stepcounter{nprop}
}

\nwc{\bgcorol}[1]{
  \settowidth{\ltheo}{Corollaire \arabic{ntheo}}
  \noindent
  \paragraph{Corollaire}% \arabic{ntheo}}
  \hspace{-0.5em}%\hspace{-0.4cm}
  \bgmp[t]{\textwidth-\ltheo-0.5em}{\it #1}\enmp
}

\newcounter{ndef}
\setcounter{ndef}{1}
\newlength{\ldef}
\nwc{\bgdef}[1]{
  \settowidth{\ldef}{Dfinition \arabic{ndef}}
  \noindent
  \paragraph{Dfinition}% \arabic{ndef}}
  \hspace{-0.5em}%\hspace{-0.4cm}
  \bgmp[t]{\textwidth-\ldef-0.5em}{\it #1}\enmp
  \stepcounter{ntheo}
}

\nwc{\bgproof}[1]{
  \vspq\noindent
  \ul{Dmonstration:} #1 
  \hfill$\square$
}

% "Cadre" type Objectifs....
\nwc{\ObjTitle}{Objectif\!\!:\ \ }
\newlength{\lgObjTitle}
\newlength{\hgObj}
\newlength{\hgObjTitle}\settoheight{\hgObjTitle}{\ObjTitle}
\newcommand{\Obj}[1]{%
  \begin{flushright}%
  \settowidth{\lgObjTitle}{\ObjTitle}
  \settototalheight{\hgObj}{\phantom{\bgmp{16.4cm}{\bf\emph{\ObjTitle}}#1\enmp}}
  \bgmp{17.1cm}
  \psline(-1ex,-\hgObj)(-1ex,-1.5\hgObjTitle)(\lgObjTitle,-1.5\hgObjTitle)\par
    \bgmp{17.cm}{\bf\emph{\ObjTitle}}#1\enmp
  \enmp
  \end{flushright}
}

\renewcommand\thesection{\Roman{section}\ \ -}
\renewcommand\thesubsection{\arabic{subsection})}
\renewcommand\thesubsubsection{\hspace*{0.5cm}\alph{subsubsection})\hspace*{-0.4cm}}

% Bandeau en bas de page
\newcommand{\TITLE}{Variables alatoires discrtes et continues}
\author{Y. Morel}
\date{}

\usepackage{fancyhdr}

\pagestyle{fancyplain}
\setlength{\headheight}{0cm}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0.5pt}
\lhead{}\chead{}\rhead{}

\lfoot{Y. Morel - \url{xymaths.free.fr/Lycee/TS/}}
\rfoot{\TITLE\ - \thepage/\pageref{LastPage}}
\cfoot{}
\title{\TITLE}

\pagestyle{fancy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}


\vspace*{-1.cm}
\ct{\LARGE{\bf \TITLE}}
\vspace{0.6cm}

\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. $X$, 
dfinie sur $[a;b]$
\[\begin{tabular}{|c|ccc|}\hline
\ \ $x$\ \  & $a$ &\hspace*{1.5cm}& $b$ \\\hline
&&&\\
$f$&\psline{->}(0.2,-0.2)(1.,0.6)&\psline{->}(0.2,0.6)(1.,-0.2)&\\
&&&\\\hline
\end{tabular}\]
%\psline(-15,-0.5)(10,-0.5)
%\psline(-15,-1.5)(10,-1.5)
\\

\vspace{-0.4cm}
\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{-0.4cm}
\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
\\

\psset{arrowsize=5pt,unit=1cm}
\begin{pspicture}(-2,-0.)(7,4.6)
  \psline{->}(-2,0)(7,0)\rput(7.3,-0.2){$x_i$}
  \psline{->}(0,-0.2)(0,4.6)\rput(-1.1,4.4){$P(X=x_i)$}
  %
  \pspolygon(-1,0)(-0.5,0)(-0.5,0.5)(-1,0.5)\rput(-0.7,-0.25){$x_1$}
  \pspolygon(-0.5,0)(-0.5,1.5)(0,1.5)(0,0)\rput(-0.2,-0.25){$x_2$}
  \pspolygon(0,3)(0,0)(.5,0)(.5,3)\rput(.3,-0.25){$x_3$}
  \pspolygon[fillstyle=solid,fillcolor=lightgray](.5,0)(.5,2.2)(1,2.2)(1,0)\rput(.8,-0.25){$x_4$}
  \psline{->}(0.75,2.22)(1.7,4)
  \rput(2.3,4.2){$p_4\!=\!P(X\!=\!x_4)$}
  %
  \rput(1.5,1.){$\dots$} 
  \rput(1.5,-0.25){$\dots$}
  %
  \pspolygon[fillstyle=vlines](2.,0)(2.,1.7)(2.5,1.7)(2.5,0)\rput(2.3,-0.25){$x_i$}
  \pspolygon[fillstyle=vlines](2.5,0)(2.5,1.2)(3,1.2)(3,0)
  %
  \rput(3.5,1.){$\dots$} 
  \rput(3.5,-0.25){$\dots$}
  %
  \pspolygon[fillstyle=vlines](4,0)(4,1.4)(4.5,1.4)(4.5,0)\rput(4.3,-0.25){$x_j$}
  %
  \rput(5.2,1.){$\dots$} 
  \rput(5.2,-0.25){$\dots$}
  %
  \pspolygon(6,0)(6,0.4)(6.5,0.4)(6.5,0)\rput(6.3,-0.25){$x_n$}
  %
  \psbrace[ref=lt,rot=270,nodesepB=-2pt](4.5,2)(2,2){}
  \rput(4.5,2.6){$P(x_i\leqslant X\leqslant x_j)=p_i+\dots+p_j$}
  %\rput(-0.25,0.5){$p_1$}
  %\rput(-0.25,1.5){$p_2$}\psline[linestyle=dotted](0,1.5)(0.5,1.5)
  %\rput(-0.25,3){$p_3$}\psline[linestyle=dotted](0,3)(1,3)
  %\rput(-0.25,2.2){$p_4$}\psline[linestyle=dotted](0,2.2)(1,2.2)
\end{pspicture}

& 
\psset{arrowsize=5pt,unit=1cm}
\begin{pspicture}(-2.5,-0.)(6,4.6)
  \psline{->}(-1.8,0)(6,0)\rput(6.3,-0.2){$x$}
  \psline{->}(0,-0.2)(0,4.6)\rput(-0.3,4.6){$f$}
  %
  \nwc{\f}[1]{2.718 -1 #1 -1 add 2 exp mul exp 4.2 mul 
    2.718 -1 #1 -2.9 add 2 exp mul exp 2 mul 
    add 0.1 add}
  \psplot[plotpoints=200]{-0.8}{4}{\f{x}}
  \psline[linestyle=dashed](-0.8,-0.1)(! -0.8 \space \f{-0.8})
  \rput(-0.8,-0.3){$a$}
  \psline[linestyle=dashed](4,-0.1)(! 4 \space \f{4})
  \rput(4,-0.25){$b$}
  % 
  \psline(1.5,0)(!1.5 \space \f{1.5})\rput(1.5,-0.3){$c$}
  \psline(2,0)(!2 \space \f{2})\rput(2,-0.25){$d$}
  \pscustom{
    \psplot{1.5}{2}{\f{x}} \gsave
    \psline(2,0)(1.5,0)
    %\fill[fillstyle=solid, fillcolor=lightgray]
    \fill[fillstyle=vlines]
    \grestore }
  \psline{->}(1.75,2.5)(2.2,3.8)
  \rput(4,4){$\dsp P(c\leqslant X \leqslant d)\!=\!\int_c^d f(x)\,dx$}
  %
  \rput(0.4,-0.28){$\alpha$}
  \psline(0.4,0)(!0.4 \space \f{0.4})
  \psline{->}(! 0.3 \space \f{0.4})(-0.5,3.2)
  \rput(-1.3,3.4){$P(X\!=\!\alpha)=0$}
\end{pspicture}


\\

\bgit
\item[$\bullet$] Esprance: $\dsp E(X)=\sum_{i=1}^n x_i\,p_i$
  \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[$\bullet$] Esprance: $\dsp E(X)=\int_{a}^{b} x\,f(x)\,dx$
  \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{Probabilits cumules 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 rpartition}
$\bgar[t]{ll}
F(x)&=P(X\!\leqslant\! x)\\[0.2cm]
&\dsp=\int_{a}^x f(t)\,dt
\enar$
\vspd

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

\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}

Haut de la page Haut de la page