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Elles sont traitées indépendamment les unes après les autres dans leur ordre d'arrivée. On suppose que le nombre de requêtes qui arrivent au serveur peut être modélisé par un processus de Poisson. Les requêtes sont envoyées vers le serveur à un} \begin{subexo}{} \end{subexo} \end{exo} diff --git a/src/serie6/exo5.tex b/src/serie6/exo5.tex index 532d17b..5479a0b 100644 --- a/src/serie6/exo5.tex +++ b/src/serie6/exo5.tex @@ -1,5 +1,18 @@ \begin{exo} - \donnee{} - \begin{subexo}{} + \donnee{Dans sa tournée, un voyageur de commerce doit se rendre de la ville A à la ville B. Il dispose de deux itinéraires : le premier en passant par la ville C et le second par la ville D. Aucune liaison directe entre A et B existe. Les temps en heures (h) que passe sur la route le voyageur de commerce pour se déplacer de A à B via les villes C et D peuvent être représentés par des variables aléatoires indépendantes $X_1,X_2,X_3,X_4$ définies par + \begin{enumerate} + \item $X_1$ : "Durée du trajet entre les villes A et C"; + \item $X_2$ : "Durée du trajet entre les villes C et B"; + \item $X_3$ : "Durée du trajet entre les villes A et D"; + \item $X_4$ : "Durée du trajet entre les villes D et B"; + \end{enumerate} +On suppose que ces variables aléatoires sont toutes issues d'une distribution normale. Les temps espérés pour se déplacer d'une ville à l'autre se trouvent dans la figure ci-dessous et le coefficient de variation de chacune de ces variables aléatoires vaut 0.2. +\begin{center}\includegraphics[scale=0.8]{ex5}\end{center} +} + \begin{subexo}{Le coefficient de variation $\delta_X$ d'une variable aléatoire X est donné par $\frac{\sigma_X}{\mu_X}$. Calculer les écarts-type des variables $X_1,X_2,X_3,X_4$.} + \end{subexo} + \begin{subexo}{Calculer la probabilité que le trajet entre la ville A et B via C dure moins de 9 heures.} + \end{subexo} + \begin{subexo}{Déterminer la probabilité que la durée du trajet entre A et B via C soit plus courte que celle via D en considérant la varible aléatoire $T = T_1 - T_2$ ou $T_1$ représente la durée du trajet via C et $T_2$ celle du trajet via D.} \end{subexo} \end{exo} diff --git a/src/serie6/exo6.tex b/src/serie6/exo6.tex index 532d17b..0c0356e 100644 --- a/src/serie6/exo6.tex +++ b/src/serie6/exo6.tex @@ -1,5 +1,7 @@ \begin{exo} - \donnee{} - \begin{subexo}{} + \donnee{Le temps en secondes que passe un internaute sur une page d'un site WEB peut être décrit par une variable aléatoire X telle que $ Y = \ln(X)$ est une variable aléatoire issue d'une distribution normale d'espérance $0.5$ et de variance $1$. On dit que X est une variable aléatoire issue d'une distribution log-normale.} + \begin{subexo}{Exprimer la fonction de répartition $F_x$ de la variable aléatoire $X$ en utilisant la fonction de répartition $\phi$.} + \end{subexo} + \begin{subexo}{Calculer la probabilité qu'une page soit regardée pendant plus de 10 secondes.} \end{subexo} \end{exo}