From cff04eca7792abf43e3578ecd1d36802cf00bcd8 Mon Sep 17 00:00:00 2001
From: ktzanev <kroum.tzanev@univ-lille.fr>
Date: Thu, 1 Feb 2024 17:59:08 +0100
Subject: [PATCH] ajustements dans fiche 4 avant impression

---
 TDs/M61B_2023-24_TD4.pdf | Bin 82756 -> 79459 bytes
 TDs/M61B_2023-24_TD4.tex | 191 +++++++++++++++++----------------------
 2 files changed, 81 insertions(+), 110 deletions(-)

diff --git a/TDs/M61B_2023-24_TD4.pdf b/TDs/M61B_2023-24_TD4.pdf
index cf8a4f6085ca9ead0523fbd40060f288f703d9b1..85f51cb80e3cf300395c2f5dc618684ec7be5004 100644
GIT binary patch
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diff --git a/TDs/M61B_2023-24_TD4.tex b/TDs/M61B_2023-24_TD4.tex
index 8263ca5..30bc459 100644
--- a/TDs/M61B_2023-24_TD4.tex
+++ b/TDs/M61B_2023-24_TD4.tex
@@ -16,19 +16,19 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
 % -----------------------------------------------
 \begin{exo}\emph{(Calculs de probabilités)}
 
-  Dans un restaurant (à l'époque ou cela existait encore), on compte 18 femmes et 12 hommes. Il y a 21 droitiers en tout, dont 15 sont des femmes. En choisissant un client au hasard, quelle est la probabilité qu'il/elle soit gaucher-ère ? Même question si l'on choisit un homme au hasard.
+  Dans un restaurant, on compte 18 femmes et 12 hommes. Il y a 21 droitiers en tout, dont 15 sont des femmes. En choisissant un client au hasard, quelle est la probabilité qu'il/elle soit gaucher-ère ? Même question si l'on choisit un homme au hasard.
 \end{exo}
 
 % -----------------------------------------------
 \begin{exo}\emph{(Les chats de Shrödinger)}
 
-  Considérons l'expérience de pensée suivante : on place 20 chats dans des boites individuelles. Par ailleurs,
+  Considérons l'expérience de pensée suivante : on place 20 chats dans des boites individuelles. Par ailleurs, en plus des chats,
   \begin{itemize}
-    \item 10 boites sont vides.
-    \item 5 boites contiennent 3g d'arsenic.
+    \item 10 boites ne contiennent rien d'autre,
+    \item 5 boites contiennent 3g d'arsenic,
     \item 5 boites contiennent 10g d'arsenic.
   \end{itemize}
-  Au bout d'une heure, la probabilité qu'un chat soit encore en vie dans la boite est égale à 1 si la boite est vide, 0,6 si la boite contient 3g d'arsenic, et 0,2 si la boite contient 10g d'arsenic\footnote{Aucun chat n'a été maltraité pour l'élaboration de cette expérience de pensée !}. On choisit une des 20 boites au hasard.
+  Au bout d'une heure, la probabilité qu'un chat soit encore en vie dans la boite est égale à 1 si la boite ne contient pas d'arsenic, 0,6 si la boite contient 3g d'arsenic, et 0,2 si la boite contient 10g d'arsenic\footnote{Aucun chat n'a été maltraité pour l'élaboration de cette expérience de pensée !}. On choisit une des 20 boites au hasard.
   \begin{enumerate}
     \item Quelle est la probabilité que le chat qui s'y trouve soit encore en vie au bout d'une heure ?
     \item Le chat est en vie. Quelle est la probabilité que la boite ait été vide quand on l'y a placé ?
@@ -39,7 +39,7 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
 % -----------------------------------------------
 \begin{exo}\emph{(Un problème de transit)}
 
-  Deux avions (A et B), contenant respectivement $n_A=20$ et $n_B=40$ passagers, atterrissent  à l'aéroport Charles de Gaulle. Dans le premier, qui vient de Séoul, en Corée du sud, chaque passager à une probabilité $ p_A=0.01$ d'être infecté par le coronavirus. Dans le second, qui vient de Houston, USA, chaque passager y a une probabilité $p_B=0.2$ d'être infecté. On suppose que les passagers sont tous  indépendants entre eux.
+  Deux avions (A et B), contenant respectivement $n_A=20$ et $n_B=40$ passagers, atterrissent  à l'aéroport Charles de Gaulle. Dans l'avion A, qui vient de Séoul (Corée du Sud), chaque passager à une probabilité $ p_A=0,01$ d'être infecté par le coronavirus. Dans le avion B, qui vient de Houston (USA), chaque passager y a une probabilité $p_B=0,2$ d'être infecté. On suppose que les passagers sont tous  indépendants entre eux.
   \begin{enumerate}
     \item On note $X_A$ et $X_B$ le nombre de passager infectés dans  chacun des avions.
     \begin{enumerate}
@@ -47,11 +47,11 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
       \item Quelle est l'espérance du nombre total de passagers infectés ?
       \item S'il y a deux ou plus passagers infectés, peu importe leur provenance, l'aéroport doit être scellé, et tous les passagers mis en quarantaine. Quelle est la probabilité que cela ne soit pas nécessaire ?
     \end{enumerate}
-    \item Un bus attend les passagers de Houston (avion B), mais un passager n'est  pas autorisé à monter si une caméra thermique lui repère de la fièvre, donc le bus ne contiendra que les $Y_B\leq 40$ passagers pour lesquels aucune fièvre n'a été repérée. La caméra repère de la fièvre pour un passager infecté dans 95\% des cas, et repère de la fièvre pour un passager sain dans 20\% des cas.
+    \item Un bus attend les passagers de Houston (avion B), mais un passager n'est pas autorisé à monter si une caméra thermique lui repère de la fièvre, donc le bus ne contiendra que les $Y_B\leq 40$ passagers pour lesquels aucune fièvre n'a été repérée. La caméra repère de la fièvre pour un passager infecté dans 95\% des cas, et repère de la fièvre pour un passager sain dans 20\% des cas.
     \begin{enumerate}
-      \item On suppose que $X_B=10$. Quelle est la distribution du nombre de passagers sains qui a pu monter dans le bus ? Quelle est la distribution  du nombre de passagers infectés qui a pu monter  dans le bus ?
-      \item On suppose toujours que $X_B=10$. On  choisit une personne au hasard dans le groupe de Houston, quelle est la probabilité $p_E$ qu'elle soit autorisée à monter dans le bus ? Même question si $X_B=k\in \{0,\dots, 40\}$.
-      \item On ne suppose plus rien sur $X_B$. On choisit au hasard un des passagers en provenance de Houston, calculer $p_E.$%=0.8-3/80
+      \item On suppose que $X_B=10$. Quelle est la distribution du nombre de passagers sains qui a pu monter dans le bus ? Quelle est la distribution du nombre de passagers infectés qui a pu monter  dans le bus ?
+      \item On suppose toujours que $X_B=10$. On choisit une personne au hasard dans le groupe de Houston, quelle est la probabilité $p_E$ qu'elle soit autorisée à monter dans le bus ? Même question si $X_B=k\in \{0,\dots, 40\}$. Quelle est la distribution de $Y_{B}$ ?
+      \item On ne suppose plus rien sur $X_B$. On choisit au hasard un des passagers en provenance de Houston, calculer $p_E$.
     \end{enumerate}
     \item On suppose que la caméra thermique a scanné également les passagers de Seoul. Quelle est la probabilité qu'elle n'ait repéré de la fièvre chez aucun des passagers des deux vols ?
   \end{enumerate}
@@ -75,25 +75,65 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
 \end{exo}
 
 
-\section{Variables aléatoires discrètes et propriétés des lois classiques}
+\section{Une variable aléatoire discrète}
+
+%-----------------------------------
+\begin{exo}\emph{(Deux dés)}
+
+  Nous lançons deux dés non truqués.
+  \begin{enumerate}
+    \item La variable aléatoire $X$ est la somme des points obtenus. Quelle est la distribution de $X$ ?
+    \item Même question pour la variable aléatoire $Y$ égale au minimum des deux résultats obtenus.
+    \item On considère la variable aléatoire $Z = (X-7)^{2}$. Quelle est la distribution de $Z$ ?
+  \end{enumerate}
+\end{exo}
+
+% -----------------------------------
+\begin{exo}\emph{(Contrôle de qualité)}
+
+  Une machine-outil produit à la chaîne des objets manufacturés et l'on sait qu'en période de marche normale la probabilité pour qu'un objet soit défectueux est $p$. On se propose de vérifier la machine. À cet effet, on définit la variable aléatoire $T_r$ égale au nombre minimum de prélèvements successifs qu'il faut effectuer pour amener $r$ objets défectueux. Calculer la loi de $T_r$.
+\end{exo}
+
+% -----------------------------------
+\begin{exo}\emph{(Mélange de lois)}
+
+  On suppose que le nombre $N$ d'œufs pondus par un insecte suit une loi de Poisson de paramètre $\alpha$:
+  \[
+    \PP{N=k}=e^{-\alpha}\frac{\alpha^k}{k\,!}\qquad k\in {\mathbb{N}}
+  \]
+  On suppose également que la probabilité de développement d'un œuf est $p$ et que les œufs sont mutuellement indépendants. On note $S$ le nombre (aléatoire) de survivants. Montrer que $S$ suit une loi de Poisson de paramètre $p\alpha$.
+\end{exo}
+
+% -----------------------------------------------
+\begin{exo}\emph{(Une autre formule pour l'espérance)}
+
+  Soit $X$ une variable aléatoire d'espace d'états $\{0,\dots, N\}$. Montrer que
+  \[
+    \EE{X}=\sum_{k=0}^{N-1}\PP{X>k}.
+  \]
+  Que peut on dire si $X$ prend ses valeurs dans $\N$ tout entier ?
+\end{exo}
+
+
+\section{Plusieurs variables aléatoires discrètes}
 
 % -----------------------------------------------
 \begin{exo}\emph{(Sur la loi uniforme)}
 
   Soit $X$, $Y$ deux variables aléatoires indépendantes suivant la loi uniforme sur $\{1,\ldots,n\}$.
   \begin{enumerate}
-    \item Déterminer $\PP{X=Y}$. %1/n
-    \item Déterminer $\PP{X\geq Y}$. %1/2+1/2n
-    \item Déterminer la loi de $X+Y$. % P(X+Y)=k=\frac{2n+1-k}{n^2}
+    \item Déterminer $\PP{X=Y}$.
+    \item Déterminer $\PP{X\geq Y}$.
+    \item Déterminer la loi de $X+Y$.
   \end{enumerate}
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{(D'après CCP 1998)}
+\begin{exo}\emph{(Loi jointe)}
 
   On considère deux variables $X$ et $Y$ à valeurs dans $\N$, et on suppose que l'on a, pour tout $(j,k)\in \N^2$
   \[
-    \PP{X=j, Y=k}=\frac{(j+k)(1/2)^{j+k}}{ej!k!}.
+    \PP{X=j, Y=k}=\frac{(j+k)(1/2)^{j+k}}{j!k!e}.
   \]
   Cette quantité est la \emph{loi jointe} du couple de variables aléatoires $(X,Y)$, dont $X$et $Y$ sont les \emph{marginales}.
   \begin{enumerate}
@@ -103,11 +143,11 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Variables de Poisson)}
 
   On considère deux variables $X$ et $Y$ à valeurs dans $\N$, et on suppose que l'on a, pour tout $(j,k)\in \N^2$
   \[
-    \PP{X=i, Y=j}=\frac{\alpha}{j!k!}.
+    \PP{X=j, Y=k}=\frac{\alpha}{j!k!}.
   \]
   \begin{enumerate}
     \item Déterminer le réel $\alpha$.
@@ -125,7 +165,7 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
     \item Soit $i\in\{1,\dots, N\}$ et $n\in \N^*$, déterminer $\PP{X_i\leq n}$ puis $\PP{X_i>n}$.
     \item On définit la V.A. $Y$ par $Y=\min_{1\leq i\leq N}X_i$, c'est à dire que pour  tout $\omega\in \Omega$, $Y(\omega)=\min\{X_i(\omega), \dots ,X_N(\omega)\}$.
     \begin{enumerate}
-      \item Soit $n\in\N^*$, calculer $\PP{Y>n}$. En déduire $ \PP{Y\leq n}$, puis $\PP{Y=n}$.
+      \item Soit $n\in\N^*$, calculer $\PP{Y>n}$. En déduire $ \PP{Y\leq n}$. Quelle est la loi de $Y$ ?
       \item $Y$ admet-elle une espérance finie ? Si oui, la calculer.
     \end{enumerate}
   \end{enumerate}
@@ -149,127 +189,58 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
   \noindent On dira alors que la distribution \emph{conditionnelle} de $X$ sachant $\{Y=m\}$ est Binomiale$(m,p)$.  Déterminer la loi de $X$.
 \end{exo}
 
-% -----------------------------------------------
-\begin{exo}\emph{(Une autre formule pour l'espérance)}
-
-  Soit $X$ une variable aléatoire d'espace d'états $\{0,\dots, N\}$. Montrer que
-  \[
-    \EE{X}=\sum_{k=0}^{N-1}\PP{X>k}.
-  \]
-  Que peut on dire si $X$ prend ses valeurs dans $\N$ tout entier ?
-\end{exo}
-
-% -----------------------------------------------
-\begin{exo}\emph{(Fonctions génératrices)}
-
-  Soient $X$ et $Y$ deux variables aléatoires à valeurs dans $\N$. On appelle \emph{fonction génératrice} de $X$ la série entière
-  \[
-    G_X(t)=\sum_{n=0}^{+\infty}\PP{X=n}t^n.
-  \]
-  \begin{enumerate}
-    \item Soit $R$ le  rayon de convergence de cette série, montrer que $R \geq 1$.
-    \item À l'aide du théorème de transfert, exprimer $G_X(t)$ comme l'espérance d'une fonction de $X$ pour $t<R$.
-    \item
-    \begin{enumerate}
-      \item Justifier que $G_X$ est de classe $C^\infty$ sur $]-1,1[$.
-      \item En calculant ses premières dérivées, justifier que sa dérivée $k$-ième $G^{(k)}_X$ est donnée par
-      \[
-        G_X^{(k)} =\sum_{n=k}^{+\infty} \P(X=n)\frac{n!}{(n-k)!}t^{n-k}.
-      \]
-      \item Exprimer $G^{(k)}_X$ en fonction de la distribution de $X$.
-      \item En déduire que si $G_X=G_Y$ sur $]-1,1[$ alors $X$ et $Y$ ont même loi.
-    \end{enumerate}
-    \item
-    \begin{enumerate}
-      \item Calculer $G_X$ lorsque $X$ suit une loi de Bernoulli de paramètre $p$ puis lorsque $X$ suit une loi binomiale de paramètres $(n,p)$.
-      \item On suppose que $X$ et $Y$ sont indépendantes. Démontrer que pour tout $t\in]-1,1[$
-      \[
-        G_{X+Y}(t)=G_X(t)G_Y(t).
-      \]
-      \item Soit $X\sim \Bin{n,p}$ et $Y\sim \Bin{m,p}$ deux variables aléatoires indépendantes. En utilisant les fonctions générarices, déterminer la loi de $X+Y$. Retrouver ce résultat sans les fonctions génératrices.
-    \end{enumerate}
-    \item On suppose maintenant que $R>1$.
-    \begin{enumerate}
-      \item On suppose que $X$ est intégrable, i.e. $\sum_{n=0}^{+\infty} n \P(X=n)<+\infty$. Calculer la dérivée $G'_X$ de $G_X$ et exprimer $\E(X)$ en fonction de $G'_X$.
-      \item On suppose maintenant que $X$ est de carré intégrable, i.e. $\sum_{n=0}^{+\infty} n^2 \P(X=n)<+\infty$. Calculer $G''_X$ et montrer que
-      \[
-        \E(X^2)=G''_X(1)+G'_X(1).
-      \]
-      \item En déduire en fonction de $G_X$ l'expression de la variance de $X$.
-    \end{enumerate}
-  \end{enumerate}
-\end{exo}
-
 
 % ===============================================
 \section{Borel-Cantelli}
 % ===============================================
 
 % -----------------------------------------------
-\begin{exo}\emph{(Borel-Cantelli et les retours à l'origine)}
+\begin{exo}\emph{(Retours à l'origine)}
 
-  Soit $\suite[n\geq1]{X_n}$ une suite de v.a.r. indépendantes telles que~:
-  \[
-    P(X_i=+1)=p \qquad \qquad P(X_i=-1)=1-p=q \qquad 0<p<1\quad p\neq \frac{1}{2}.
-  \]
-  On pose:
+  Considérons un ivrogne qui, à chaque étape, fait un pas en avant avec une probabilité $p > \frac{1}{2}$ et un pas en arrière avec une probabilité $q=1-p$. Soit $\suite[n\geq1]{X_n}$ une suite de variables aléatoires indépendantes de même loi décrivant les déplacements de l'ivrogne, où
   \[
-    S_n=\sum_{i=1}^{n} X_i \qquad A_n=\{S_n=0\}.
+    P(X_i=+1)=p, \qquad P(X_i=-1)=q.
   \]
-  L'événement $A_n$ est un retour à zéro. On pose
-  \[
-    A\coloneqq
-    \ensemble{\omega\in\Omega}{\text{la suite $S_n(\omega)$ repasse une infinité de fois en $0$}}
-    = \cap_{k\geq 1}\cup_{j\geq k}A_j.
-  \]
-  Prouver que $P(A)=0$.
-
-  %En utilisant la loi forte des grands nombres, donner une
-  %conclusion plus précise permettant de retrouver le résultat précédent.
+  On pose la variable \enquote{position} $ S_n \coloneqq \sum_{i=1}^{n} X_i $ et l'événement \enquote{retour à l'origine} $ A_n \coloneqq \{S_n=0\}$.
+  Déterminer la probabilité que l'ivrogne retourne une infinité de fois à l'origine.\\
+  \emph{Rappelons la formule de Stierling $n! \sim_\infty \sqrt{2\pi n} \left(\frac{n}{e}\right)^n $.}
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Convergence presque sûre)}
 
-  Soit $\alpha>0$. Considérons une suite de variables aléatoires indépendantes $\suite[n\geq 1]{X_n}$ telles que
-  \[
-    P(X_n=1)=\frac {1}{n^{\alpha}}=1-P(X_n=0).
-  \]
+  Soit $\alpha>0$. Considérons une suite de variables aléatoires indépendantes de Bernoulli $\suite[n\geq 1]{X_n}$ de paramètres $p_{n}=n^{-\alpha}$, c.-à-d. $X_{n}\sim \mathcal{B}(p_n)$ pour tout $n\geq 1$.
   \begin{enumerate}
-    \item Montrer que $\lim_{n\to +\infty}E(X_n)=0$.
-    \item Comme on le verra plus en détail dans le cours, on dit que $(X_n)_{n\in \N}$ converge presque surement vers une variable aléatoire $X$ si $ \PP{\lim_{n\to\infty} X_n=X}=1$. Étudier la convergence presque sûre de la suite $\suite[n\geq 1]{X_n}$.
+    \item Soit $L$ l'événement \enquote{La suite $\suite[n\geq 1]{X_n}$ admet une limite}. Calculer $\PP{L}$ en fonction de $\alpha$.
+    \item On dit que $(X_n)_{n\in \N}$ converge presque sûrement vers une variable aléatoire $X$ si
+    \[
+      \PP{\lim_{n\to\infty} X_n=X}=1.
+    \]
+    Discuter la convergence presque sûre de la suite $\suite[n\geq 1]{X_n}$ en fonction de $\alpha$.
   \end{enumerate}
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Désaccord avec Borel ?)}
 
-  Considérons un jeu infini de pile ou face avec une pièce équilibrée et définissons la suite de variables aléatoires $\suite[k\geq 1]{Y_k}$ de la manière suivante :
+  Considérons un jeu infini de pile ou face avec une pièce équilibrée et définissons la suite d'événements
   \[
-    Y_k=\left\{
-      \begin{array}{cl}
-        0 & \textrm{ si le premier jet donne face}\\
-        \left\{\begin{array}{ll}
-          0 & \textrm{ si le k-ème jet donne face}\\
-          1 & \textrm{ si le k-ème jet donne pile}.
-        \end{array}\right. & \textrm{ sinon.}
-      \end{array}
-      \right.
+    F_{n} : \text{\enquote{le premier et le $k$-ème jet donne face}}.
   \]
-  Soit $E_n$ l'événement $\{Y_n=1\}$ pour $n\in\N^*$ et $E$ l'événement \enquote{les événements $E_n$ se produisent infiniment souvent}.
+  pour $n\in\N^*$ et $F=\limsup F_{n}$ l'événement \enquote{les événements $F_n$ se produisent infiniment souvent}.
   \begin{enumerate}
     \item Montrer que
     \[
-      \sum_{n\in\N^*}P(E_n)=+\infty\quad \textrm{et}\quad P(E)=\frac 12.
+      \sum_{n\in\N^*}P(F_n)=+\infty\quad \textrm{et}\quad P(F)=\frac{1}{2}.
     \]
-    \item Expliquer pourquoi ceci n'est pas en contradiction avec le second Lemme de Borel-Cantelli.
+    \item Expliquer pourquoi ceci n'est en contradiction ni avec le second lemme de Borel-Cantelli, ni avec la loi du \enquote{zéro-un} de Borel.
   \end{enumerate}
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{(L'armée des singes dactylographes)}
+\begin{exo}[.7]\emph{(Le paradoxe du singe savant)}
 
-  Montrer que dans le jeu de pile ou face infini, la séquence \texttt{pfffp} apparaît presque sûrement une infinité de fois. Généraliser.
+  Un singe (immortel) tape indéfiniment et au hasard sur le clavier d'une machine à écrire. Montrer que presque sûrement le singe écrira un nombre infini de fois l'énoncé de ce même exercice.
 \end{exo}
 
 \end{document}
-- 
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