From 5e2cec0743b5aa72518dca9ea01d4d7036478616 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Wed, 28 Oct 2020 11:42:57 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=20bilan=20de=20l'=C3=A9tape=201?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../01_Binomiale_et_echantillonnage/1B.tex | 14 ---- .../1B_bernoulli_binomiale.pdf | Bin 0 -> 49459 bytes .../1B_bernoulli_binomiale.tex | 70 ++++++++++++++++++ .../01_Binomiale_et_echantillonnage/index.rst | 8 ++ 4 files changed, 78 insertions(+), 14 deletions(-) delete mode 100644 Complementaire/01_Binomiale_et_echantillonnage/1B.tex create mode 100644 Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.pdf create mode 100644 Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.tex diff --git a/Complementaire/01_Binomiale_et_echantillonnage/1B.tex b/Complementaire/01_Binomiale_et_echantillonnage/1B.tex deleted file mode 100644 index 18a3b8c..0000000 --- a/Complementaire/01_Binomiale_et_echantillonnage/1B.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass[a4paper,10pt]{article} -\usepackage{myXsim} - -\author{Benjamin Bertrand} -\title{Binomiale et echantillonnage - Cours} -\date{octobre 2020} - -\pagestyle{empty} - -\begin{document} - -\maketitle - -\end{document} \ No newline at end of file diff --git a/Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.pdf b/Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4b5bc692db53e8d6bcbe7837e68b3836e6e06d46 GIT binary patch literal 49459 zcmce-Q?M}2wk5i3+xGgFZGX$QZQHhO+qP}nwr%ylPoERFBl_X?%ibAP8HrgLHEPV7 zQ8Qx{shsd%8U|VxDAKv5;Wa2`0(t^l0}Ci_ZYVk>cROPOIyG4X3u8kkC^|)F1E>EQ z`D^Q74Mis?XzNCxN&k;M1O0!U+E8@=n)$D*7@+8+^_?8d|FK~B$Ks#)f7Sn3C>T51 zIy)E|I})(`Gok=RCuMA7>SRX1z(Vlf<)2l|t(=S<2;VCQe*@k@8QU2B_sl=7f89FUKMVh7Ah7?tAaMN0ATa!koPP@q z4F971Kf?n9yFOu>R*j{L6Fy 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z8l2f#d&<+rof{3W)P=HGXIGi1?8I4f6pX?TICApz*Z+ef{@(;gO~J?*noiu>(8v{E zlOCU)O&gj{(ahcGUk{p266<~4upc4522BdAm#@==evN6 zTe~cq@hC#Azg{_Xe5a<~+hF56)OZ>gUfHUWp72&Hw$x b931uR9bNwsaA;-@dU_^k5)xrq5$OK`OEK9n literal 0 HcmV?d00001 diff --git a/Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.tex b/Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.tex new file mode 100644 index 0000000..c4b2d90 --- /dev/null +++ b/Complementaire/01_Binomiale_et_echantillonnage/1B_bernoulli_binomiale.tex @@ -0,0 +1,70 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Binomiale et echantillonnage - Cours} +\date{octobre 2020} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\section{Expérience et loi de Bernoulli} + +\subsection*{Définition} + +Une expérience aléatoire qui a deux issues possibles (que l'on nommera \textbf{succès} et \textbf{échec}) est appelé \textbf{expérience de Bernoulli}. + +En associant la valeur 1 à un succès et 0 à un échec. On peut modéliser cette expérience avec un variable aléatoire $X$ qui suit un \textbf{loi de Bernoulli} (notée $X \sim \mathcal{B}(p)$) résumée par le tableau suivant: + +\begin{center} + \begin{tabular}{|c|*{2}{C{2cm}|}} + \hline + Valeurs & 1 & 0 \\ + \hline + Probabilité & p & 1-p \\ + \hline + \end{tabular} +\end{center} + +où $p$ est la probabilité d'avoir un succès. + +\subsubsection*{Exemple} +Un passager qui a 9 chances sur 10 de se présenter à l'embarquement d'un avion. + +\afaire{Préciser ce qu'est le succès, l'échec, déterminer la valeur de $p$ et compléter le tableau} + +\subsection*{Propriétés} +Soit $X \sim \mathcal{B}$ alors +\begin{itemize} + \item L'espérance de $X$ est $E[X] = p$ + \item L'écart-type de $X$ est $\sigma = p(1-p)$ +\end{itemize} + +\subsubsection*{Démonstration} +\afaire{Démontrer la formule de l'espérance} + +\section{Loi binomiale} + +On a vu que pour simuler tout un vol, c'est à dire 53 passagers, il fallait répéter 53 fois la loi de Bernoulli vue dans l'exemple précédent. Les répétitions de loi de Bernoulli s'appellent \textbf{schéma de Bernoulli} et sont modéliser avec une loi \textbf{binomiale}. + +\subsection*{Définition} + +La \textbf{loi Binomiale de paramètre $n$ et $p$} notée $\mathcal{B}(n;p)$ est la loi de probabilité qui modélise la répétition indépendantes et identiques de $n$ situations modélisées par une loi de Bernoulli de paramètre $p$. + +\bigskip + +Ces situations peuvent être représenté par un arbre de probabilité où chaque étage correspond à une répétition. + +\subsubsection*{Exemple} + +Dans une classe de 20 élèves, Sarah ne veut pas être interrogée sur son travail. Le professeur interroge au hasard 3 élèves qu'il choisit de façon indépendantes et identiques. + +On note $X$ le nombre de fois que Sarah est interrogée. + +\afaire{Quelle loi suit $X$? Représenter la situation avec un arbre de probabilité} + + +\end{document} diff --git a/Complementaire/01_Binomiale_et_echantillonnage/index.rst b/Complementaire/01_Binomiale_et_echantillonnage/index.rst index 470f1f2..fbdfbae 100644 --- a/Complementaire/01_Binomiale_et_echantillonnage/index.rst +++ b/Complementaire/01_Binomiale_et_echantillonnage/index.rst @@ -13,8 +13,16 @@ Binomiale et echantillonnage Activité avec le tableur où l'on essaie de simuler une situation de suréservation d'un avion. +.. image:: ./1E_surreservation.pdf + :height: 200px + :alt: Simulation avec le tableur de la surréservation d'avions. + Bilan: définitions de loi de Bernoulli et de la loi binomiale (caractères pour modéliser et représentation par un arbre). +.. image:: ./1B_bernoulli_binomiale.pdf + :height: 200px + :alt: Cours sur la loi de bernoulli et la loi binomiale. + Étape 2: Étude de situations aléatoires et répétées ===================================================