From dd1fad59663af9be95d0d72946fb46ec2263e2f5 Mon Sep 17 00:00:00 2001
From: ktzanev <kroum.tzanev@univ-lille.fr>
Date: Sun, 10 Mar 2024 18:15:01 +0100
Subject: [PATCH] rajout du DS1 de 9 mars 2024

---
 Exam/M61B_2023-24_DS1-solutions.pdf | Bin 0 -> 77388 bytes
 Exam/M61B_2023-24_DS1-sujet.pdf     | Bin 0 -> 47694 bytes
 Exam/M61B_2023-24_DS1.tex           | 230 ++++++++++++++++++
 Exam/images/attention.pdf           | Bin 0 -> 2229 bytes
 Exam/images/attention.tex           |   4 +
 Exam/images/lille.pdf               | Bin 0 -> 7797 bytes
 Exam/images/tsvp.pdf                | Bin 0 -> 2580 bytes
 Exam/images/tsvp.tex                |   7 +
 Exam/lille.sty                      | 361 ++++++++++++++++++++++++++++
 Exam/m61proba.tex                   |  66 +++++
 README.md                           |   6 +
 11 files changed, 674 insertions(+)
 create mode 100644 Exam/M61B_2023-24_DS1-solutions.pdf
 create mode 100644 Exam/M61B_2023-24_DS1-sujet.pdf
 create mode 100644 Exam/M61B_2023-24_DS1.tex
 create mode 100644 Exam/images/attention.pdf
 create mode 100644 Exam/images/attention.tex
 create mode 100644 Exam/images/lille.pdf
 create mode 100644 Exam/images/tsvp.pdf
 create mode 100644 Exam/images/tsvp.tex
 create mode 100644 Exam/lille.sty
 create mode 100644 Exam/m61proba.tex

diff --git a/Exam/M61B_2023-24_DS1-solutions.pdf b/Exam/M61B_2023-24_DS1-solutions.pdf
new file mode 100644
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diff --git a/Exam/M61B_2023-24_DS1.tex b/Exam/M61B_2023-24_DS1.tex
new file mode 100644
index 0000000..0e87ad8
--- /dev/null
+++ b/Exam/M61B_2023-24_DS1.tex
@@ -0,0 +1,230 @@
+\documentclass[a4paper,12pt,reqno]{amsart}
+\usepackage{lille}
+\lilleset{
+  solutions,
+  % sans enonces,
+  titre=\sisujet{Devoir surveillé}\sisolutions{Solutions du devoir surveillé},
+  date=9 mars 2024,
+  duree=2 heures,
+}
+\input{m61proba}
+
+\begin{document}
+\sisujet{
+  \tsvp
+  \vspace{7mm}
+  \attention~
+  \emph{Les documents et les objets électroniques sont interdits. Les exercices sont indépendants. Toutes les réponses doivent être justifiées.}
+  \vspace{7mm}
+}
+
+% -----------------------------------------------
+\begin{exo}
+
+  Sur l'ensemble $\Omega=\{ -2, -1, 0, 1, 2 \}$ on note $\mathcal{F}$ la
+  plus petite tribu qui contient $\{ -2, -1, 0 \}$ et $\{ 0, 1, 2 \}$.
+  \begin{enumerate}
+    \item Expliciter $\mathcal{F}$ en donnant la liste de ses éléments (sans justification).
+    \item On munit $\R$ de sa tribu borélienne. Parmi ces trois fonctions de
+    $\Omega$ dans $\R$, lesquelles sont $\mathcal{F}$-mesurables ? Justifier vos réponses.
+    \begin{enumerate}
+      \item la fonction nulle : $\forall \omega\in\Omega$, $f_0(\omega)=0$;
+      \item la fonction identité : $\forall \omega\in\Omega$, $f_1(\omega)=\omega$;
+      \item la fonction signe : $\forall \omega\in\Omega$, $f_2(\omega)=-1$ si $\omega < 0$ et $f_2(\omega)=1$ si $\omega\geq 0$.
+    \end{enumerate}
+  \end{enumerate}
+\end{exo}
+
+\begin{solution}
+  \begin{enumerate}
+    \item $\mathcal{F} =\ens[\big]{\emptyset, \ens{-2,-1}, \ens{0},\ens{1,2}, \ens{-2,-1, 0}, \ens{0, -2,-1}, \ens{-2,-1, 1, 2}, \Omega}$.
+
+    \emph{Justification non exigée.} Montrons que $\mathcal{F}$ est la tribu engendrée par $A_1$ et $A_2$. Il est facile de vérifier que $\mathcal{F}$ est stable par complémentaire et union (dénombrable), donc $\mathcal{F}$ est une tribu qui contient $A_1 = \ens{-2,-1, 0}$ et $A_2 = \ens{0, 1, 2}$. Ainsi il reste à montrer que les quatre autres ensembles non triviaux sont engendrés par $A_1$ et $A_2$ :
+    $\ens{-2,-1} = A_1\cap\overline{A_2}$, $\ens{1,2} = A_2\cap\overline{A_1}$, $\ens{0} = A_1\cap A_2$ et $\ens{-2,-1,1,2} = \overline{A_1\cap A_2}$.
+    \begin{enumerate}
+      \item La fonction nulle est mesurable, car elle est constante. Toute fonction constante est mesurable, car l'image réciproque de tout ensemble (borélien) est $\Omega$ ou $\emptyset$.
+      \item La fonction identité n'est pas mesurable, car l'image réciproque de $\{1\}\in\mathcal{B}(\mathbb{R})$ est $\{1\}$ qui n'est pas dans $\mathcal{F}$.
+      \item La fonction signe est mesurable car l'image réciproque de tout ensemble (borélien) $A$ est dans $\mathcal{F}$:
+      \[
+        f_2^{-1}(A) = \begin{cases}
+          \emptyset\in\mathcal{F} & \text{si } -1\notin A \text{ et } 1\notin A,\\
+          \{-2,-1\}\in\mathcal{F} & \text{si } -1\in A \text{ et } 1\notin A,\\
+          \{0, 1,2\}\in\mathcal{F} & \text{si } -1\notin A \text{ et } 1\in A,\\
+          \Omega\in\mathcal{F} & \text{si } -1\in A \text{ et } 1\in A.
+        \end{cases}
+      \]
+    \end{enumerate}
+  \end{enumerate}
+\end{solution}
+
+% -----------------------------------------------
+\begin{exo}
+
+  On note $\lambda$ la mesure de Lebesgue sur $\R$.
+  \begin{enumerate}
+    \item Soit $U$ un ouvert de $\R$. Montrer que si $U$ est borné alors $\lambda(U)<+\infty$.\\ La réciproque est-elle vraie ?
+    \emph{Vous pouvez considérer des ouverts centrés en $n$ pour $n\in\N$.}
+    \item Soit $\varepsilon>0$. Construire un ouvert $U$ dense dans $\R$ de sorte que $\lambda(U)\leq\varepsilon$.\\
+    \emph{Vous pouvez considérer des ouverts centrés en $q$ pour $q\in\Q$.}
+    \item Soit $A$ un borélien de $\R$. Montrer que si $A$ contient un ouvert non vide, alors $\lambda(A)>0$.\\
+    Si $B=[0,1]\setminus\Q$, que vaut $\lambda(B)$ ? $B$ peut-il contenir un ouvert non vide ?
+  \end{enumerate}
+\end{exo}
+
+\begin{solution}
+  \emph{(Il s'agit de l'exercice 8 de la feuille 3, déjà fait en TD.)}
+  \begin{enumerate}
+    \item Soit $U$ un ouvert de $\R$. Si $U$ est borné alors il existe $M>0$ tel que $U \subset[-M,M]$. Par la monotonie de la mesure on obtient $\lambda(U)\leq \lambda([-M,M]) = 2M <+\infty$.\\
+    La réciproque n'est pas vraie, car il existe des ouverts de mesure finie qui ne sont pas bornés. Par exemple, $U=\bigcup_{n\geq 1} ]n-2^{-n},n+2^{-n}[$ est un ouvert de mesure finie $\lambda(U)\leq \sum_{n\geq 1} \lambda\bpa{]n-2^{-n},n+2^{-n}[} = \sum_{n\geq 1} 2^{-n+1} = 2$, mais n'est pas borné, car $\mathbb{N}^*$ est inclus dans $U$.
+    \item Soit $\varepsilon>0$. Comme $\mathbb{Q}$ est dénombrable, ordonnons-le dans une suite $\suiteN{q_n}$. Soit $U = \bigcup_{n\geq 0} ]q_n-\varepsilon\,2^{-n-2},q_n+\varepsilon\,2^{-n-2}[$. Alors $U$ est un ouvert dense dans $\R$, car il contient $\mathbb{Q}$, et $\lambda(U) \leq \sum_{n\geq 0} \lambda\bpa{]q_n-\varepsilon\,2^{-n-2},q_n+\varepsilon\,2^{-n-2}[} = \sum_{n\geq 0} \varepsilon\, 2^{-n-1} = \varepsilon$.
+    \item Soit $A$ un borélien de $\R$. Si $A$ contient $O$ un ouvert non vide, alors il existe $x\in O$ et $\varepsilon>0$, tels que $]x-\varepsilon,x+\varepsilon[\subset O\subset A$. Donc $0 < 2\varepsilon \leq \lambda(O)\leq\lambda(A)$.\\
+    Vu que $\lambda(\mathbb{Q}\cap[0,1]) = 0$, comme ensemble dénombrable, nous avons $\lambda(B) = 1 -0 = 1$, mais $B$ ne peut pas contenir un ouvert non vide, car $\mathbb{Q}$ rencontre tout ouvert non vide. \emph{Ainsi la réciproque \enquote{un borélien de mesure positive contient un ouvert non vide} est fausse.}
+  \end{enumerate}
+\end{solution}
+
+\sisujet{\newpage}
+% -----------------------------------------------
+\begin{exo}
+
+  Soient deux réels $\mu>0$ et $p\in]0,1[$. Dans une banque, le nombre de chèques émis par les clients en un jour est une variable aléatoire $X$ qui suit la loi de Poisson $\mathcal{P}(\mu)$. Pour chaque chèque émis, la probabilité que ce chèque soit sans provision est $p$.
+
+  On appelle $Y$ le nombre de chèques émis sans provision lors d'une journée.
+  \begin{enumerate}
+    \item Soit $n \in \N$. Trouver $\PP{Y=k \mid X=n}$ et $\PP{X-Y=l \mid X=n}$ suivant les valeurs de $k\in\N$ et $l\in\N$.
+    \item Utiliser la formule des probabilités totales pour déterminer la loi de $Y$ et celle de $X-Y$.
+    \item Calculer $\PP{X-Y=l \mid Y=k}$ en fonction de $k$ et $l$. Les variables $X-Y$ et $Y$ sont-elles indépendantes ?
+  \end{enumerate}
+\end{exo}
+
+\begin{solution}
+  \emph{(Cet exercice est fortement inspiré de l'exercice 12 de la feuille 4, fait en TD.)}
+  \begin{enumerate}
+    \item L'entier $n$ est fixé. On sait que $n$ chèques ont été émis aujourd'hui. On doit trouver la probabilité que $k$ chèques soient sans provision. Elle est nulle si $k>n$ puisqu'il ne peut pas y avoir plus de chèques sans provision que de chèques en tout :
+    \[
+      \PP{Y=k \mid X=n} = 0 \quad\text{ si }\quad k>n.
+    \]
+    Si $0\le k \le n$ la probabilité que $k$ chèques soient sans provision est la probabilité qu'il y ait $k$ succès (le \enquote{succès} est le fait d'être sans provision) parmi $n$ expériences (chaque chèque représente une expérience aléatoire) qui ont toutes la même probabilité de succès $p$ (la probabilité pour un chèque d'être sans provision). Si on suppose les chèques indépendants, ce qui est raisonnable, on est exactement dans la situation d'un schéma de Bernoulli : la probabilité qu'il y ait $k$ succès est donc
+    \[
+      \PP{Y=k\mid X=n} = \binom{n}{k} p^k (1-p)^{n-k} \quad\text{ si }\quad 0\leq k\leq n.
+    \]
+    Pour trouver la loi du nombre $X-Y$ de chèques avec provision sachant qu'il y a $X=n$ chèques émis, on fait de même en remplaçant la probabilité $p$ qu'un chèque soit sans provision par la probabilité $q = 1-p$ qu'un chèque soit avec provision. Ainsi
+    \[
+      \PP{X-Y=l\mid X=n} = \binom{n}{k} (1-p)^l p^{l-k} \quad\text{ si }\quad 0\leq k\leq l.
+    \]
+    \item Il peut y avoir $0$ ou $1$ ou $2$ ou\dots  chèques sans provision, donc l'ensemble des valeurs que $Y$ peut prendre est $Y(\Omega)=\N$. Déterminer la loi de $Y$ consiste à déterminer $\PP{Y=k}$ pour chaque valeur $k$ que peut prendre $Y$. On fixe un $k$ (réécrire avec $k=5$ si on a besoin d'une valeur numérique pour bien voir que $k$ est fixé) et on utilise la formule de conditionnement par tous les cas possibles :
+    \begin{align*}
+      \PP{Y=k}
+        & = \sum_{n=0}^{+\infty} \PP{Y=k\mid X=n} ~\PP{X=n} \\
+      \intertext{\emph{(pour $n<k$ on a vu que la proba conditionnelle est nulle)}}
+        & = \sum_{n=k}^{+\infty} \PP{Y=k\mid X=n} ~\PP{X=n} \\
+      \intertext{\emph{(pour $n \ge k$ la proba conditionnelle a une forme binomiale)}}
+        & = \sum_{n=k}^{+\infty} \binom{n}{k} p^k (1-p)^{n-k} ~\PP{X=n} \\
+        & = \sum_{n=k}^{+\infty} \frac{n!}{k!~(n-k)!} ~ p^k (1-p)^{n-k} ~\PP{X=n}.
+    \end{align*}
+    La variable $X$ qui suit la loi de Poisson $\mathcal{P}(\lambda)$ signifie que $\PP{X=n}=e^{-\lambda}\frac{\lambda^n}{n!}$ pour tout $n\in\N$, donc
+    \begin{align*}
+      \PP{Y=k}
+        & = \sum_{n=k}^{+\infty} \frac{n!}{k!~(n-k)!} ~ p^k (1-p)^{n-k} ~e^{-\lambda}\frac{\lambda^n}{n!}\\
+      \intertext{\emph{(on simplifie par $n!$ et on sort ce qui ne dépend pas de l'indice de sommation $n$)}}
+        & = \frac{1}{k!}~ p^k ~ e^{-\lambda} \sum_{n=k}^{+\infty} \frac{1}{(n-k)!} ~ (1-p)^{n-k} ~\lambda^n.
+    \end{align*}
+    Pour que la somme démarre à $0$, on fait le changement d'indice en $i=n-k$ donc $n=i+k$ :
+    \begin{align*}
+      \PP{Y=k}
+        & = \frac{1}{k!}~ p^k ~ e^{-\lambda} \sum_{i=0}^{+\infty} \frac{1}{i!} ~ (1-p)^{i} ~\lambda^{i+k} \\
+      \intertext{\emph{(évidemment $\lambda^{i+k}=\lambda^i \lambda^k$)}}
+        & = \frac{1}{k!}~ p^k ~ \lambda^{k} ~ e^{-\lambda} \sum_{i=0}^{+\infty} \frac{1}{i!} ~ (1-p)^{i} ~\lambda^{i} \\
+      \intertext{\emph{(tout aussi évidemment $\lambda^k p^k = (\lambda p)^k$ et pareil pour $(\lambda(1-p))^i$)}}
+        & = \frac{(\lambda p)^k}{k!} ~ e^{-\lambda} \sum_{i=0}^{+\infty} \frac{(\lambda(1-p))^i}{i!} \\
+      \intertext{\emph{(à ce stade on reconnaît la série exponentielle et c'est gagné)}}
+        & = \frac{(\lambda p)^k}{k!} ~ e^{-\lambda} e^{\lambda(1-p)} = \frac{(\lambda p)^k}{k!} ~ e^{-\lambda p}
+    \end{align*}
+    \emph{Moralité : le nombre $Y$ de chèques sans provision suit la loi de Poisson de paramètre $\lambda p$.}
+
+    Pour trouver la loi du nombre $X-Y$ de chèques avec provision, on peut faire la même chose : écrire les probabilités conditionnelles des $X-Y=l$ sachant $X=n$, calculer $\PP{X-Y=l}$ par la formule de conditionnement par tous les cas possibles, enlever les termes nuls, etc. On s'aperçoit très vite que ça revient à refaire exactement tout ce qui a été fait ci-dessus en remplaçant la probabilité $p$ qu'un chèque soit sans provision par la probabilité $1-p$ qu'un chèque soit avec provision. Sans surprise, après le même calcul avec $1-p$ au lieu de $p$, on trouve la même loi avec $1-p$ au lieu de $p$.
+
+    \emph{Moralité : Le nombre $X-Y$ de chèques avec provision suit la loi de Poisson de paramètre $\lambda(1-p)$.}
+    \item Les variables $X-Y$ et $Y$ sont-elles indépendantes ?
+
+    L'argument \enquote{$X-Y$ n'est pas indépendant de $Y$ parce que dans l'écriture de $X-Y$ il y a $Y$} ne résiste pas au lavage : la définition de l'indépendance dans le cours n'est pas basée sur le nom des v.a. La définition, c'est que $X-Y$ et $Y$ sont indépendantes si
+    \[
+      \forall k\in\N, \quad \forall l\in\N, \quad \PP{\{Y=k\}\cap\{X-Y=l\}}=\PP{Y=k}~\PP{X-Y=l}.
+    \]
+    On fixe un $k$ et un $l$ dans $\N$ et on regarde si l'égalité a lieu. C'est pareil d'avoir $k$ chèques sans provision et $l$ chèques avec provision, ou $k$ chèques sans provision et $k+l$ chèques en tout donc :
+    \begin{align*}
+      \PP{\{Y=k\}\cap\{X-Y=l\}} & = \PP{\{Y=k\}\cap\{X=k+l\}}      \\
+                                & = \PP{Y=k\mid X=k+l}~\PP{X=k+l}.
+    \end{align*}
+    Comme $k+l \ge k$, ça donne
+    \begin{align*}
+      \PP{\{Y=k\}\cap\{X-Y=l\}}
+        & = \binom{k+l}{k} p^k (1-p)^{k+l-k} ~e^{-\lambda}\frac{\lambda^{k+l}}{(k+l)!} \\
+      \intertext{\emph{(le $(k+l)!$ est au numérateur et au dénominateur, on simplifie)}}
+        & = \frac{1}{k!~l!} p^k (1-p)^l~ e^{-\lambda}~\lambda^{k+l}
+    \end{align*}
+    Il faut regarder si ceci est égal à $\PP{Y=k}~\PP{X-Y=l}$. On sait que $\PP{X-Y=l}=\frac{(\lambda(1-p))^l}{l!} ~ e^{-\lambda(1-p)}$ puisque $X-Y$ suit la loi de Poisson de paramètre $\lambda(1-p)$. Ainsi
+    \begin{align*}
+      \PP{Y=k}~\PP{X-Y=l}
+        & = \frac{(\lambda p)^k}{k!} ~ e^{-\lambda p} ~ \frac{(\lambda(1-p))^l}{l!} ~ e^{-\lambda(1-p)} \\
+        \intertext{\emph{(on simplifie et on regroupe)}}
+        & = \frac{\lambda^{k+l} ~p^k ~ (1-p))^l}{k!~l!} ~ e^{-\lambda}.
+    \end{align*}
+    On constate que la probabilité de l'intersection et le produit des probabilités sont égales pour tout $k$ et $l$ de $\N$, donc $X-Y$ et $Y$ sont indépendantes !
+
+    \emph{Est-ce étonnant ? Ça signifie que le nombre de chèques avec provision faits chaque jour est indépendant du nombre de chèques sans provision faits chaque jour. Comme les clients de la banque agissent indépendamment sans tenir compte des chèques rédigés par les autres clients, c'est normal que le nombre de chèques avec provision soit indépendant du nombre de chèques sans provision.}
+  \end{enumerate}
+\end{solution}
+
+% -----------------------------------------------
+\begin{exo}
+
+  Dans tout l'exercice, $X$ désigne une variable aléatoire à valeurs dans $]0,1[$ et de loi uniforme sur $]0,1[$. On pose
+  \[
+    Z\coloneqq\frac{1-X}{X}.
+  \]
+  \begin{enumerate}
+    \item Calculer explicitement la fonction de répartition de la variable aléatoire positive $Z$.\\ Dessiner son graphe.
+    \item La loi de $Z$ est-elle à densité ? Si oui, la calculer et dessiner son graphe.
+    \item Expliquer, si possible sans calcul, pourquoi $Z$ et $1/Z$ ont \emph{même loi}.
+    \item On brise une tige de longueur $1$ en choisissant au hasard le point de rupture suivant une loi uniforme sur $]0,1[$. Quelle est la probabilité que l'un des deux morceaux soit plus de deux fois plus long que l'autre ?\\
+    \indication{Vous pouvez faire cette question en utilisant les questions précédentes avec $X$ la longueur du morceau gauche, mais vous pouvez aussi la faire directement.}
+  \end{enumerate}
+\end{exo}
+
+\begin{solution}
+
+  On note $F_Z$ la fonction de répartition de $Z$.
+  \begin{enumerate}
+    \item On remarque que $Z$ prend des valeurs positives donc si $z<0$, $F(z)=0$. Soit donc $z\geq 0$.
+    Comme $X \geq 0$ et $1 + z > 0$ pour $z\geq 0$, nous avons $F_Z(z) = \PP{\frac{1-X}{X}\leq z} = \PP{1\leq X(1+z)} = \PP{X \geq \frac{1}{1+z}} = 1-F_{X}(\frac{1}{1+z})$. Et comme $\frac{1}{1+z}\in]0,1]$ pour $z\geq0$, on trouve $F_Z(z) = 1-\frac{1}{1+z} = \frac{z}{1+z}$.
+    Et donc finalement
+    \[
+      F_Z(z) =
+      \begin{cases}
+        0             & \text{si } z <0,    \\
+        \frac{z}{1+z} & \text{si } z\geq 0.
+      \end{cases}
+    \]
+
+    \item La fonction $F_Z$ est continue, de classe $C^1$ par morceaux, donc $Z$ est à densité $f = F'_{Z}$ sur $\R^{*}$, ainsi, en choisissant arbitrairement la valeur en $0$, on trouve une densité
+    \[
+    f(z) =
+    \begin{cases}
+      0                 & \text{si } z \leq 0,  \\
+      \frac{1}{(1+z)^2} & \text{si } z > 0.
+    \end{cases}
+    \]
+
+    \item On a $\frac{1}{Z} = \frac{X}{1-X} =  \frac{1-(1-X)}{1-X}$. Or, $1-X$ est aussi de loi uniforme sur $]0,1[$, donc $Z$ et $\frac{1}{Z}$ ont même loi.
+
+    \item Soit $X$ le point de rupture, alors les deux morceaux sont de longueur $X$ et $1-X$. La condition \enquote{l'un des deux morceaux est plus de deux fois plus long que l'autre} se traduit par la réunion des deux conditions incompatibles : $X\geq 2(1-X)$ ou $(1-X) \geq 2X$.
+
+    \emph{Méthode I.} On cherche $\PP{(X\geq 2(1-X)) \sqcup (1-X \geq 2X)} =\PP{\frac{1-X}{X}\leq \frac{1}{2}} + \PP{\frac{X}{1-X}\leq \frac{1}{2}} = 2 \PP{Z\leq \frac{1}{2}}$, puisque, d'après la question précédente, $Z$ et $\frac{1}{Z}$ ont même loi. Ainsi la probabilité que l'un des deux morceaux soit plus de deux fois plus long que l'autre est $2F_Z(\frac{1}{2}) = \frac{2}{3}$.
+
+    \emph{Méthode II.} On peut aussi voir, sans utiliser $Z$, que $X\geq 2(1-X) \iff 3X\geq 2 \iff X\geq\frac{2}{3}$ et $1-X\geq 2X \iff 3X\leq 1 \iff X\leq\frac{1}{3}$. Ainsi la probabilité recherchée est $\PP{X\geq\frac{2}{3}} + \PP{X\leq\frac{1}{3}} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$.
+  \end{enumerate}
+\end{solution}
+
+
+\end{document}
+
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diff --git a/Exam/images/attention.tex b/Exam/images/attention.tex
new file mode 100644
index 0000000..4a5f0bb
--- /dev/null
+++ b/Exam/images/attention.tex
@@ -0,0 +1,4 @@
+\documentclass[tikz]{standalone}
+\begin{document}
+  \tikz{\node(e){!};\draw[red,ultra thick,line join=round,yshift=-.15ex](90:1em)--(210:1em)--(330:1em)--cycle;}
+\end{document}
diff --git a/Exam/images/lille.pdf b/Exam/images/lille.pdf
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diff --git a/Exam/images/tsvp.tex b/Exam/images/tsvp.tex
new file mode 100644
index 0000000..86631f1
--- /dev/null
+++ b/Exam/images/tsvp.tex
@@ -0,0 +1,7 @@
+\documentclass[tikz]{standalone}
+\usetikzlibrary{shapes.arrows}
+\begin{document}
+  \begin{tikzpicture}
+    \node[single arrow, fill=black!3, draw=black, inner sep=3mm]{\textsf{ T.S.V.P.}};
+  \end{tikzpicture}
+\end{document}
diff --git a/Exam/lille.sty b/Exam/lille.sty
new file mode 100644
index 0000000..d5dc78c
--- /dev/null
+++ b/Exam/lille.sty
@@ -0,0 +1,361 @@
+\ProvidesPackage{lille}
+%-------------- les options
+\usepackage{pgfopts}
+\def\lilleset{\pgfqkeys{/lille}}% ≈ \def\lilleset#1{\pgfkeys{/lille/.cd,#1}}
+\newif\iflille@sans@logo% par défaut c'est avec logo
+\newif\iflille@solutions% par défaut les solution sont cachées
+  \newcommand{\sisolutions}[1]{\iflille@solutions #1\fi}% à aficher si l'option 'solutions' est présente
+  \newcommand{\sisujet}[1]{\iflille@solutions\else #1\fi}% à aficher si l'option 'solutions' n'est présente
+\newif\iflille@sans@enonces% par défaut les énoncés sont affichés
+\lilleset{
+    couleur exo/.store in=\lille@exo@color,% la couleur pour les traits à côté des exercices
+    couleurexo/.store in=\lille@exo@color,% for package options (spaces are lost)
+    couleur sol/.store in=\lille@sol@color,% la couleur pour les traits à côté des solutions
+    couleursol/.store in=\lille@sol@color,% for package options (spaces are lost)
+    couleur liens/.store in=\lille@link@color,% la couleur pour les liens
+    couleurliens/.store in=\lille@link@color,% for package options (spaces are lost)
+    couleur/.style = {% pour définir les trois couleurs en même temps
+      couleur exo=#1,
+      couleur sol=#1!49!black,
+      couleur liens=#1
+    },
+    couleur=blue,% la couleur par défaut
+    sans logo/.is if=lille@sans@logo,% cacher le logo de l'université
+    sanslogo/.is if=lille@sans@logo,% for package options (spaces are lost)
+    solutions/.is if=lille@solutions,% afficher les solutions
+    sans enonces/.is if=lille@sans@enonces,% cacher les énoncés (si on veut juste les solutions)
+    sansenonces/.is if=lille@sans@enonces,% for package options (spaces are lost)
+    titre/.store in=\lille@titre,% le titre de la feuille ou du sujet d'examen
+    formation/.store in=\lille@formation,% la formation (ex: Master 1\iere année)
+    annee/.store in=\lille@annee,% l'année (ex: 2015--2016)
+    parcours/.store in=\lille@parcours,% le parcours (ex: parcours Mathématiques)
+    module/.store in=\lille@module,% le module (ex: Algèbre 1)
+    date/.store in=\lille@date,% la date de l'examen
+    duree/.store in=\lille@duree,% la durée de l'examen
+}
+\ProcessPgfPackageOptions*
+% ------------- encodage
+\usepackage{iftex}
+\ifPDFTeX % PDFLaTeX
+  \usepackage[utf8]{inputenc}
+  \usepackage[T1]{fontenc}
+  \usepackage{textcomp} % pour utiliser certains caractères
+  \usepackage{lmodern}
+\else % LuaLaTeX & XeLaTeX
+  \usepackage{fontspec}
+\fi
+% ------------- géométrie de la page
+\usepackage[top=21mm,left=23mm,right=21mm,bottom=21mm,nohead,nomarginpar,includeall,columnsep=14mm]{geometry}
+% pour espacer un peut le texte
+\parskip=1mm
+\parindent=0mm
+\let\blskip\baselineskip
+\renewcommand{\baselinestretch}{1.3}
+\setlength{\abovedisplayskip}{2pt plus 4pt}
+\setlength{\belowdisplayskip}{3pt plus 4pt}
+\setlength{\abovedisplayshortskip}{1pt plus 4pt}
+\setlength{\belowdisplayshortskip}{2pt plus 4pt}
+% ------------- les standards
+\usepackage{mathtools,amssymb,bm,bbm,amsthm}
+\usepackage[french]{babel}
+\usepackage[babel=true]{csquotes} % pour utiliser \enquote
+\usepackage{xspace} % pour ne pas «gobble» des espaces après \ieme par exemple
+% ------------- autres symboles
+% Pour éviter d'utiliser les packages pas très compatibles avec amssymb.
+% Déclare \llbracket and \rrbracket à partir de `fourier`
+\DeclareFontEncoding{FMS}{}{}
+\DeclareFontSubstitution{FMS}{futm}{m}{n}
+\DeclareSymbolFont{fourier}{FMS}{futm}{m}{n}
+\DeclareFontEncoding{FMX}{}{}
+\DeclareFontSubstitution{FMX}{futm}{m}{n}
+\DeclareSymbolFont{largefourier}{FMX}{futm}{m}{n}
+\DeclareMathDelimiter{\llbracket}{\mathopen}{fourier}{153}{largefourier}{133}
+\DeclareMathDelimiter{\rrbracket}{\mathclose}{fourier}{154}{largefourier}{134}
+% Décalre \wideparen à partir de `yhmath`
+\DeclareFontFamily{OMX}{yhex}{}
+\DeclareFontShape{OMX}{yhex}{m}{n}{<->yhcmex10}{}
+\DeclareSymbolFont{yhlargesymbols}{OMX}{yhex}{m}{n}
+\DeclareMathAccent{\wideparen}{\mathord}{yhlargesymbols}{"F3}
+% \leq et \req à la française (\oleq et \ogeq sont les originaux)
+\let\oleq\leq
+\let\ogeq\geq
+\let\leq\leqslant
+\let\geq\geqslant
+% ------------- Pour ajuster les enumerate
+\usepackage[inline,shortlabels]{enumitem}
+% compatibilité enumitem - franch babel (d'après la doc de 28 septembre 2014, p.14)
+\newlength\mylabelwidth
+\newcommand*{\mylabel}{$\triangleright$}% ou \textendash (tiret plus court)
+\settowidth{\mylabelwidth}{\mylabel}
+\setlist[itemize]{label=\mylabel, nosep}
+% la numérotation des questions
+\setlist[1]{
+  labelindent=\parindent,
+  listparindent=\parindent,
+  label={\textbf{\alph{enumi})}},
+  ref={\alph{enumi}}
+}
+\setlist[2]{
+  label={(\roman{enumii})},
+  ref={\alph{enumi}.\roman{enumii}}
+}
+\setlist{labelwidth=\mylabelwidth,
+        itemsep=0.4ex plus 0.2ex minus 0.2ex,
+        parsep=0.4ex plus 0.2ex minus 0.2ex,
+        topsep=0.8ex plus 0.4ex minus 0.4ex,
+        partopsep=0.4ex plus 0.2ex minus 0.2ex,
+        leftmargin=!}
+
+% mon changement perso
+\setlist{leftmargin=7mm}
+% ------------- Pour prévenir les coupures des pages au mauvaise endroits
+\usepackage{needspace}
+% ------------- Pour l'affichage conditionnel des solutions et des énoncés
+\usepackage{environ}
+\long\def\environment@body#1{\relax}% the body collected by \Collect@Body in \hide@body is ignored
+\def\hide@body{\expandafter\Collect@Body\expandafter\environment@body}% hide the environment body
+% ------------- Pour les couleurs
+\usepackage{xcolor}
+
+%       Macros
+% ---------------------
+
+% sections
+\renewcommand{\section}[1]{%
+  \needspace{7\baselineskip}%
+  \vspace{4mm}\hfill%
+  \setlength{\fboxsep}{2.1mm}%
+  \fcolorbox{\lille@exo@color}{\lille@exo@color!3}{%
+    \parbox{\dimexpr\columnwidth-21mm\relax}{%
+      \centering\vspace{1pt}%
+      \textsf{\large #1}%
+    }%
+  }%
+  \hfill\vspace*{3mm}%
+}
+
+% subsections
+\renewcommand{\subsection}[1]{%
+  \needspace{7\baselineskip}%
+  \vspace{3mm}\hfill%
+  \setlength{\fboxsep}{1.4mm}%
+  \fcolorbox{\lille@exo@color}{\lille@exo@color!2}{%
+    \parbox{\dimexpr\columnwidth-42mm\relax}{%
+      \centering\vspace{1pt}%
+      \textsf{\large #1}%
+    }%
+  }%
+  \hfill\vspace*{2mm}%
+}
+
+% Les numéros d'exercices
+\newcounter{numeroexo}
+\setcounter{numeroexo}{0}
+% Les lignes à côté de l'exercice
+\newlength{\img@vone}\setlength{\img@vone}{2.1pt}% la hauteur de la première ligne
+\newlength{\img@vtwo}\setlength{\img@vtwo}{1.4pt}% la hauteur de la deuxième ligne
+\newcommand\heightratio[1]{% pour changer la hauteur des lignes à côté de l'exercice (coefficiant multiplicateur)
+  \edef\vlineone{\strip@pt\dimexpr 0.4pt + #1\img@vone\relax}
+  \edef\vlinetwo{\strip@pt\dimexpr 0.4pt + #1\img@vtwo\relax}
+}
+\heightratio{1}% par défaut 1
+\newcommand*\exoimg[2]{% #1 : la couleur, #2 : la largeur de la ligne horizontale
+\setlength{\unitlength}{1cm}%
+\begin{picture}(0,0)(0,0)
+  \color{#1}
+  \put(-0.1,0.4){\line(0,-1){\vlineone}}
+  \put(-0.14,0.4){\line(0,-1){\vlinetwo}}
+  \put(-0.4,-0.1){\line(1,0){#2}}
+\end{picture}%
+}
+% Le macro principal pour les exercices/problèmes (à cahcher si l'option 'sans enonces' est présente)
+\newlength{\afterexo}\setlength{\afterexo}{3mm plus 4mm}% l'espace après les exercices, les problèmes et les solutions
+\newenvironment{exercice}[1]% macro principal pour les exercices/problèmes
+  {\needspace{3\baselineskip}\vspace{1mm}\par\refstepcounter{numeroexo}\hspace*{-1mm}%
+   \noindent{\exoimg{\lille@exo@color}{2.5}\bfseries\sffamily #1 \arabic{numeroexo}}}%
+  {\par\vspace{\afterexo}}
+% --- exo → Exercice (à afficher seulement si l'option 'sans enonces' n'est pas présente)
+\newenvironment{exo}[1][1]%
+  {%
+    \iflille@sans@enonces%
+      \hide@body
+    \else%
+      \setcounter{equation}{0}%
+      \heightratio{#1}%
+      \begin{exercice}{Exercice} \normalfont%
+    \fi%
+  }%
+  {\unless\iflille@sans@enonces\end{exercice}\fi}
+
+% --- bleme → Problème (à afficher seulement si l'option 'sans enonces' n'est pas présente)
+\newenvironment{bleme}[1][1]%
+  {%
+    \iflille@sans@enonces%
+      \hide@body%
+    \else%
+      \setcounter{equation}{0}%
+      \heightratio{#1}%
+      \begin{exercice}{Problème} \normalfont%
+    \fi%
+  }%
+  {\unless\iflille@sans@enonces\end{exercice}\fi}
+
+% exo difficile (avec étoile)
+\newcommand{\hard}{\raisebox{.7ex}{\llap{${}^{\star}$\kern 2pt}}}
+
+% Les solutions (à afficher seulement si l'option 'solutions' est présente)
+\def\numsolution{% le numéro de la solution (à afficher seulement si l'option 'sans enonces' est présente)
+  \iflille@sans@enonces%
+    \refstepcounter{numeroexo}~\arabic{numeroexo}%
+  \else%
+    ~:
+  \fi%
+}
+\newenvironment{solution}% affichage de la solution si l'option 'solutions' est présente
+  {%
+    \iflille@solutions
+      \needspace{2\baselineskip}\vspace{1mm}\par\hspace*{-2.1mm}%
+      \noindent{
+        \exoimg{\lille@sol@color}{2.59}%
+        \bfseries\sffamily Solution\numsolution%
+      }%
+    \else%
+      \hide@body%
+    \fi%
+  }%
+  {\iflille@solutions\par\vspace{\afterexo}\fi}
+
+% -------------- Pour les figures
+\usepackage{graphicx}
+  \graphicspath{{images/}} % pour chercher dans le sous-dossier images
+\usepackage[export]{adjustbox} % pour pouvoir utiliser valign=t, vspace, margin, ...
+
+% Les images à mettre à droite ou à gauche
+\newsavebox{\imager@box}% used to recover the image width
+\def\imager@set{\pgfqkeys{/imager}}% used to set options
+\imager@set{
+  % toutes les clés inconnues sont stockées dans \imager@options et transmises à \includegraphics (avec adjustbox)
+  .unknown/.code={
+    \edef\imager@options{\imager@options,\pgfkeyscurrentname=#1}
+  },
+  options/.estore in=\imager@options,
+  % le décalage de l'image en points
+  x offset/.store in=\imager@dx,
+  y offset/.store in=\imager@dy,
+  offset/.style 2 args={x offset=#1, y offset=#2},% pour définir les deux décalages en même temps
+  % la longueur de séparation entre l'image et le texte
+  sep/.store in=\imager@sep,
+}
+% ---- imageR ------
+% Place l'image à droite et le texte à gauche
+% Example : \imageR[width=49mm, sep=7mm, y offset=17, trim={28 0 7 0}]{image.pdf}{text}
+%   les options 'sep=7mm' et 'y offset=17' sont spécifiques à \imageR et \imageL
+%   les autres : 'width=49mm' et 'trim={28 0 7 0}' sont passées à \includegraphics (avec adjustbox)
+\newcommand{\imageR}[3][]{%
+\imager@set{
+  options={valign=t},% pour que l'image soit alignée en haut (option de adjustbox)
+  offset={0}{0},% pas de décalage par défaut
+  sep={7pt},% séparation entre l'image et le texte par défaut
+  #1% les options passées à \imageR par le paramètre optionnel
+}%
+\sbox{\imager@box}{\expandafter\includegraphics\expandafter[\imager@options]{#2}}% enregistre l'image dans une boîte pour récupérer sa largeur
+\vskip 0pt%
+\noindent%
+\begin{minipage}[t]{\dimexpr \textwidth - \wd\imager@box - \imager@sep}% la boîte de texte
+  #3
+\end{minipage}%
+\hfill%
+\begin{minipage}[t]{\wd\imager@box}% la boîte de l'image
+  \begin{picture}(0,0)
+    \put(\imager@dx,\imager@dy){\usebox{\imager@box}}% on affiche l'image avec le décalage
+  \end{picture}
+\end{minipage}%
+}
+% ---- imageL ------
+% Place l'image à gauche et le texte à droite
+% Example : \imageL[width=49mm, sep=7mm, y offset=17, trim={28 0 7 0}]{image.pdf}{text}
+\newcommand{\imageL}[3][]{%
+\imager@set{
+  options={valign=t},% pour que l'image soit alignée en haut (option de adjustbox)
+  offset={0}{0},% pas de décalage par défaut
+  sep={7pt},% séparation entre l'image et le texte par défaut
+  #1% les options passées à \imageR par le paramètre optionnel
+}%
+\sbox{\imager@box}{\expandafter\includegraphics\expandafter[\imager@options]{#2}}% enregistre l'image dans une boîte pour récupérer sa largeur
+\vskip 0pt%
+\noindent%
+\begin{minipage}[t]{\wd\imager@box}% la boîte de l'image
+  \begin{picture}(0,0)
+    \put(\imager@dx,\imager@dy){\usebox{\imager@box}}% on affiche l'image avec le décalage
+  \end{picture}
+\end{minipage}%
+\hfill%
+\begin{minipage}[t]{\dimexpr \textwidth - \wd\imager@box - \imager@sep}% la boîte de texte
+  #3
+\end{minipage}%
+}
+
+% ---- haut de la page ------
+\if@twocolumn
+  \usepackage{cuted}% pour afficher le haut de la page sur toute la largeur en cas de deux colonnes
+\fi
+\newcommand{\lille@sidef}[2][]{% affiche #1#2 si #2 est défini (ex: \lille@sidef[\sc]{lille@formation})
+  \ifcsname#2\endcsname%
+    {#1\csname #2\endcsname\relax}%
+  \fi%
+}
+\newcommand{\lille@sipar}[1]{% insère fin de paragraph si #1 est défini (ex: \lille@sipar{lille@date})
+  \ifcsname#1\endcsname%
+    \par%
+  \fi%
+}
+\AtBeginDocument{% le code à exécuter au début du document (pour afficher le haut de la page entre autres)
+  % pas de numéros des pages
+  \pagestyle{empty}
+  \thispagestyle{empty}
+  % même si deux colonnes, le haut est sur toute la largeur de la page
+  \if@twocolumn\begin{strip}\fi
+  % afficher le logo de l'université
+  \unless\iflille@sans@logo
+    \vspace*{-2cm}
+    \begin{center}
+      \includegraphics[width=7cm]{lille.pdf}
+    \end{center}
+    \vspace*{-4mm}
+  \fi
+  \lille@sidef[\sc]{lille@formation}\hspace*{\fill}\lille@sidef[\sc\large]{lille@annee}\\
+  \lille@sidef[\sc]{lille@parcours}\hspace*{\fill}\lille@sidef[\sc\large]{lille@module}
+
+  \rule[0.5ex]{\textwidth}{0.1mm}
+  \vskip 3mm
+  \begin{center}
+      {\lille@sidef[\sc\Large]{lille@titre}\lille@sipar{lille@date}
+      \lille@sidef{lille@date}\lille@sipar{lille@duree}
+      \def\dureepar#1\relax{[ durée : #1 ]}
+      \lille@sidef[\dureepar]{lille@duree}
+      }
+  \end{center}
+  \rule[0.5ex]{\textwidth}{0.1mm}
+  \if@twocolumn\end{strip}\fi
+}
+
+% Macros pour les sujets d'examen
+
+% ---- Attention ------
+% \newcommand{\attention}{\raisebox{1.8pt}[0pt][0pt]{\textcolor{red}{\fontencoding{U}\fontfamily{futs}\selectfont\char 66\relax}}} % ne fonctionne plus, affiche un autre caractère
+\newcommand{\attention}{\raisebox{-3.5pt}[0pt][0pt]{\includegraphics{attention.pdf}}}
+
+% ---- TSVP ------
+\usepackage[absolute,overlay]{textpos}
+  \TPGrid[21mm,21mm]{1}{1} % the margins are set to 21mm and then the width and the height are diveded to 1 step, so (1,1) is at botom righ corner (minus the margin)
+\newcommand{\tsvp}{%
+  \begin{textblock}{2}[1,0](1,1)%
+    \hfill%
+    \includegraphics[scale=.7]{tsvp.pdf}%
+  \end{textblock}%
+}
+
+
+% ---- hyperref (dernier) ----
+\usepackage[pdfencoding=auto,colorlinks=true,allcolors=\lille@link@color]{hyperref}% pour les liens. En règle générale il doit être le dernier package chargé.
+
diff --git a/Exam/m61proba.tex b/Exam/m61proba.tex
new file mode 100644
index 0000000..7c0b55e
--- /dev/null
+++ b/Exam/m61proba.tex
@@ -0,0 +1,66 @@
+%==============================================================
+\lilleset{
+  formation=Licence 3\ieme année,
+  parcours=parcours Mathématiques,
+  annee=2023--2024,
+  module={M61B, Probabilités et intégration},
+  couleur=orange,
+}
+%==============================================================
+% -------------- Autres bibliothèques
+\usepackage{dsfont} % pour \mathds{1}
+\usepackage{mathrsfs} % pour \mathscr{ }
+
+% -------------- Les abréviations pour M1AN
+% les ensembles standard
+\def\R{\mathbb{R}}
+\def\C{\mathbb{C}}
+\def\N{\mathbb{N}}
+\def\Z{\mathbb{Z}}
+\def\Q{\mathbb{Q}}
+
+% Macros Clément Erignoux
+\newcommand{\pa}[1]{\left(#1 \right)}
+\newcommand{\bpa}[1]{\bigl(#1 \bigr)}
+\newcommand{\1}{\mathds{1}}
+\newcommand{\un}[1]{\mathbf{1}_{\{#1\}}}
+
+% Macros Kroum
+\newcommand*{\suite}[2][n]{\left( #2 \right)_{#1}}% exemple \suite{u_n}, \suite[k>0]{u_k}
+\newcommand*{\suiteN}[1]{\suite[n\in\N]{#1}}% exemple \suiteN{u_n}
+\newcommand*{\abs}[1]{\left\lvert{\ifx\hfuzz#1\hfuzz \,\cdot\,\else#1\fi}\right\rvert}% |.|
+\newcommand*{\norm}[1]{\left\lVert{\ifx\hfuzz#1\hfuzz \,\cdot\,\else#1\fi}\right\rVert}% norme
+\newcommand*{\scalprod}[3][]{#1\langle{#2}\kern1pt #1|{#3}#1\rangle}% exemple \scalprod[\big]{A}{B}
+\newcommand*{\ens}[2][]{#1\{ #2 #1\}}% exemple \ens[\big]{A,B,C}
+\newcommand*{\ensemble}[3][]{#1\{ #2 \;#1|\; #3 #1\}}% exemple \ensemble[\big]{x^2}{x \in \R}
+\newcommand{\iintv}[1]{\ldbrack #1\rdbrack}% intervalle d'entiers
+\newcommand{\dd}{\,\mathrm{d}}% le «d» du dx, dt, ...
+
+% les fractions comme \sfrac
+\usepackage{xfrac}
+\newcommand*{\dsfrac}[2]{\raisebox{-.1em}{\scalebox{1.4}{\sfrac{#1}{#2}}}} % bigger \sfrac
+
+% probabilité, moyenne
+\renewcommand{\P}{\mathbb{P}}
+\newcommand{\PP}[1]{\P\left( #1 \right)}
+\newcommand{\E}{{\mathbb{E}}}
+\newcommand{\EE}[1]{\E\left( #1 \right)}
+\newcommand{\V}{\mathbb{V}}
+\newcommand{\VV}[1]{\V\left( #1 \right)}
+
+% lois
+\newcommand{\Ber}[1]{\operatorname{Ber}(#1)}% loi de Bernoulli
+\newcommand{\Bin}[1]{\operatorname{Bin}(#1)}% loi binomiale
+\newcommand{\Exp}[1]{\mathcal{E}(#1)}% loi exponentielle
+\newcommand{\Uni}[1]{\mathcal{U}(#1)}% loi uniforme
+
+% indications
+\newcommand{\indication}[1]{\emph{Indication : #1}}
+
+% pour surligner
+\sisolutions{
+  \usepackage{soul}
+  \colorlet{hl}{yellow!35!white}
+  \sethlcolor{hl}
+  \renewcommand{\hl}[1]{\relax\ifmmode\colorbox{hl}{\ensuremath{#1}}\else\texthl{#1}\fi}
+}
diff --git a/README.md b/README.md
index 95038c6..a27c6f0 100644
--- a/README.md
+++ b/README.md
@@ -16,12 +16,18 @@ Dans [ce dépôt](https://gitlab.univ-lille.fr/tzanev/l3m61proba/) vous pouvez t
 - TD n°7 **[[pdf](TDs/M61B_2023-24_TD7.pdf)]** [[tex](TDs/M61B_2023-24_TD7.tex)]
 - TD n°8 **[[pdf](TDs/M61B_2023-24_TD8.pdf)]** [[tex](TDs/M61B_2023-24_TD8.tex)]
 
+### Les sujets d'examens et leurs solutions
+
+- DS1 **[[sujet](Exam/M61B_2023-24_DS1-sujet.pdf)]** **[[solutions](Exam/M61B_2023-24_DS1-solutions.pdf)]** [[tex](Exam/M61B_2023-24_DS1.tex)]
+
 ### Source complémentaires
 
 Pour compiler ces fichiers vous avez besoin des styles et des logos suivants :
 - [lille.sty](TDs/lille.sty)
 - [m61proba.tex](TDs/m61proba.tex)
 - [lille.pdf](TDs/images/lille.pdf)
+- [tsvp.pdf](Exam/images/tsvp.pdf)
+- [attention.pdf](Exam/images/attention.pdf)
 
 ### Récupération
 
-- 
GitLab