From 1cf23200793e3346c12fa4bc41c6d1649080add5 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Fri, 8 May 2020 09:19:30 +0200 Subject: [PATCH] Feat: fin du chapitre sur la loi exponentielle --- .../Loi_exponentielle/1E_decouverte.tex | 23 +- .../Loi_exponentielle/2E_loi_exp.tex | 63 +--- .../Loi_exponentielle/3E_proba_esp.pdf | Bin 32289 -> 33041 bytes .../Loi_exponentielle/3E_proba_esp.tex | 37 +-- .../Loi_exponentielle/4E_annales.pdf | Bin 0 -> 107350 bytes .../Loi_exponentielle/4E_annales.tex | 20 ++ .../Probabilite/Loi_exponentielle/banque.tex | 279 ++++++++++++++++++ .../Probabilite/Loi_exponentielle/index.rst | 8 +- Tsti2d/semaines.rst | 10 +- 9 files changed, 334 insertions(+), 106 deletions(-) create mode 100644 Tsti2d/Probabilite/Loi_exponentielle/4E_annales.pdf create mode 100644 Tsti2d/Probabilite/Loi_exponentielle/4E_annales.tex create mode 100644 Tsti2d/Probabilite/Loi_exponentielle/banque.tex diff --git a/Tsti2d/Probabilite/Loi_exponentielle/1E_decouverte.tex b/Tsti2d/Probabilite/Loi_exponentielle/1E_decouverte.tex index af64db2..971ac52 100644 --- a/Tsti2d/Probabilite/Loi_exponentielle/1E_decouverte.tex +++ b/Tsti2d/Probabilite/Loi_exponentielle/1E_decouverte.tex @@ -7,23 +7,14 @@ \pagestyle{empty} +\DeclareExerciseCollection{banque} +\xsimsetup{ + step={1}, +} + \begin{document} -\begin{exercise}[subtitle={Pannes}] - Une entreprise fabriquant des téléviseurs a effectué un suivi de la première panne des appareils qu'elle a fabriqués et vendus. - - On a réalisé ci-dessous un histogramme résumant les résultats (on a porté en abscisses la durée en mois et en ordonnées la fréquence). Les classes ont une amplitude de 1 mois. - - \includegraphics[scale=0.6]{./fig/graph_panne} - - Par exemple, 1,5\% des appareils vendus ont subi leur première panne 16 mois après leur achat par le client. - - \begin{enumerate} - \item Quelle est la fréquence des appareils qui ont subit leur première panne dès le premier mois? - \item Quelle est la fréquence des appareils qui ont subit leur première panne au 36e mois? - \item Quelle est la fréquence des appareils qui ont subit leur première panne avant le 16e mois? 36e mois? - \item Représenter sur le graphique ces deux quantités. - \end{enumerate} -\end{exercise} +\input{banque.tex} +\printcollection{banque} \end{document} diff --git a/Tsti2d/Probabilite/Loi_exponentielle/2E_loi_exp.tex b/Tsti2d/Probabilite/Loi_exponentielle/2E_loi_exp.tex index 78e92f5..fe836f2 100644 --- a/Tsti2d/Probabilite/Loi_exponentielle/2E_loi_exp.tex +++ b/Tsti2d/Probabilite/Loi_exponentielle/2E_loi_exp.tex @@ -7,63 +7,14 @@ \pagestyle{empty} +\DeclareExerciseCollection{banque} +\xsimsetup{ + step={2}, +} + \begin{document} -\begin{exercise}[subtitle={Pannes}] - On reprend l'activité commencée précédemment. Cette fois-ci, on modélise temps avant la première panne par une loi exponentielle de paramètres 0.02. - - On note $X$ la variable qui représenter le temps avant la première panne. On a donc $X \sim \mathcal{E} (0.02)$ et le loi de densité est - \begin{enumerate} - \item Quelle est la formule de la densité, $f(x)$, de $X$? - \item Démontrer que $F(x) = -e^{-0.02x}$ est une primitive de $f(x)$. - \item Calculer les quantités suivantes - \[ - \int_0^{16} f(x) \; dx \qquad \qquad - \int_0^{36} f(x) \; dx - \] - \item En déduire les quantités - \[ - P(X \leq 16) \qquad \qquad P(X\leq 36) - \] - \item Calculer les probabilités suivantes - \[ - P(X \leq 24) \qquad \qquad P(X \leq 12) - \] - \item Quelle est la probabilité que l'appareil tombe en panne la première année? - \item Quelle est la probabilité que l'appareil tombe en panne la deuxième année? - \item Quelle est la probabilité que l'appareil tombe en panne après la fin de la 3e année? - \end{enumerate} -\end{exercise} - -\begin{exercise}[subtitle={Loi exponentielle}] - \begin{enumerate} - \item Soit $X$ une variable aléatoire qui suit une loi exponentielle de paramètre 0.5. - \begin{enumerate} - \item Quelle est la densité de $x$? On la notera $f(x)$. - \item Démontrer qu'une primitive de $f(x)$ est $F(x) = -e^{-0.5x}$ - \item Calculer les probabilités suivantes - \[ - P(X < 1) \qquad P(X < 10) \qquad P(1 < X < 2) - \] - \end{enumerate} - \item Soit $Y \sim \mathcal{E}(0.01)$, calculer les quantités suivantes - \[ - P(Y < 1) \qquad P(Y < 10) \qquad P(10 < Y < 20) - \] - \item Soit $Z \sim \mathcal{E}(0.9)$, calculer les quantités suivantes - \[ - P(Z < 1) \qquad P(Z < 0.2) \qquad P(0.5 < Z < 0.6) - \] - \end{enumerate} -\end{exercise} - -\begin{exercise}[subtitle={à l'envers}] - Soit $T \sim \mathcal{E}(0.02)$. - - Déterminer $x$ tel que $P(T \leq x) = 0.5$. - - Comment interpréter le résultat dans le cadre du premier exercice? - -\end{exercise} +\input{banque.tex} +\printcollection{banque} \end{document} diff --git a/Tsti2d/Probabilite/Loi_exponentielle/3E_proba_esp.pdf b/Tsti2d/Probabilite/Loi_exponentielle/3E_proba_esp.pdf index ffe90597e497364081d6a44d3f8a71b7ff662bfe..18a7f38b742a99c141efc2ea5ff7aac13a3026fe 100644 GIT binary patch delta 8599 zcmai)RZtuNlx1-zxI4iiG}etZ5}e@f?(P<(g9X=y5S-xdF2UU`xJz(%-OSF;?#E8m zd_Aptf46R(dcR&^6N2EXxsiyts5z*eOl{G?U^IZdg`<_bH8m$E2naPnV1k^c$~mn_ zVShV+B8W1Sr)*wg^jeaerc<2C5sla6U*ey_&BV=-NyT5g*|2KH`jT(WuQJ#734zLU zoa7A5|4hPg@R9c5Hac2MUQ&#YRvS&$WI9?d);3;I?#)}sS2$b9mmVb-pHY>Ir&BV> zWRWS;_%nVQU|PfZk=YzPbOebnHBJEZ@D25ZGm}nT^uAyc|6m~`orvMzbLk`&c^jBRkFwDv2cAA~%4-$VXDUw$@(+B==n&LdOf#Xm z;kZa~@E@o5m}X!5nr*d#eb4#!gd04ILC+VVv-(Jm%HW)f7+H)IJOdGUZA9P={V-yS zr;1B#l--f*{#UY+6Avzy2QXGKuxZT{e0N3LfV^Jhk+qI*Kq!HmFPW0Yqu$Ms^f_7F z3uTrl$7f+*(?;WyU^t*vW zL%g1q5nV@lqaS4U^ZpRRG|!GoQmdWH%n>Pk?sA&dp~l`{`9pMfIyTFz-cmu0V<&~h zu5H7@6=k@7!*#jNwFP<$Hkp;w_O(k4(Fh25V)Ts;dyz5345)nb{v>nD`5$A-WN>lY z?V2oW2ly@U)|Nh{ukwhc9IvOmvMEb($R&y^{Ms7LG2BY|#fk&5AZ692BNS+4v7cEZ zvZ=8~|H<%YvfL$tbElwH*6GUh*m+0DGyQDIKSg+j^o)6KTlh3!1&DkZLGd@WPrY9p zxe8Sy&r1q1EEDTN%kJ2!?PZS9dv16|bwj;iO}^Cs9;%2?Z zy+j~-SWis(`Cu62`DFv07vS_t;*r(R0H~n$Eb5f|2KWy9M#**PcE&iyM`)BI*`W8! 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zWulcGFfia%9L diff --git a/Tsti2d/Probabilite/Loi_exponentielle/3E_proba_esp.tex b/Tsti2d/Probabilite/Loi_exponentielle/3E_proba_esp.tex index ed6ba53..e236fb8 100644 --- a/Tsti2d/Probabilite/Loi_exponentielle/3E_proba_esp.tex +++ b/Tsti2d/Probabilite/Loi_exponentielle/3E_proba_esp.tex @@ -7,37 +7,14 @@ \pagestyle{empty} +\DeclareExerciseCollection{banque} +\xsimsetup{ + step={3}, +} + \begin{document} -\begin{exercise}[subtitle={Durée de vie}] - On note $T$ la variable aléatoire qui modélise la durée de fonctionnement d'un tube fluorescent. On suppose que $T$ suit une loi exponentielle de paramètre 0.0015. - - \begin{enumerate} - \item Quelle est la densité de $T$? - \item Calculer les probabilités des évènements suivants - \begin{itemize} - \item A: "la durée de bon fonctionnement est compris entre 600h et 700h" - \item B: "la durée de bon fonctionnement est inférieur à 800h" - \item C: "Le tube fonctionne encore après 750h" - \item D: "Le tube fonctionne a arrêté de fonctionner à l'instant 750h" - \end{itemize} - \item Calculer l'espérance de $T$. Interpréter le résultat. - \end{enumerate} -\end{exercise} - -\begin{exercise}[subtitle={Durée de vie - encore}] - On note $T$ la variable aléatoire qui modélise la durée de fonctionnement d'un composant électronique. On suppose que $T$ suit une loi exponentielle dont on ignore le paramètre. - - \begin{enumerate} - \item Une étude a montré qu'en moyenne la durée de fonctionnement de ce composant est de 5ans. En déduire le paramètre de la loi. - \item Quelle est la densité de $T$? - \item Calculer les probabilités des évènements suivants - \begin{itemize} - \item A: "la durée de bon fonctionnement est compris entre 1 et 2ans" - \item B: "la durée de bon fonctionnement est inférieur à 3ans" - \item C: "Le tube fonctionne encore après 10ans" - \end{itemize} - \end{enumerate} -\end{exercise} +\input{banque.tex} +\printcollection{banque} \end{document} diff --git a/Tsti2d/Probabilite/Loi_exponentielle/4E_annales.pdf b/Tsti2d/Probabilite/Loi_exponentielle/4E_annales.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0135d86ae8856207425a1106cecca7a4216e657 GIT binary patch literal 107350 zcmce-Wo#zB)+PFcnVHjJX6AI5nL5nO%*@Qp++pTCVTKMfbJAhvdCzy}ySi61(#)T8 zELF*6@3OqKcI~pgY;r|Wae8J3c3AR1i^Hq1Y{X2&_C{8)e0;Ess-6y}#EhB>MpmZA zF0hO$u0}5ZRU&ThWDCpq^QXNBu{P5`_RLKGb?L$~{;TG{-eQJjlrwa3vi!$_`5%ja z>i>2B$3ofE+1}O3*wmSr{htzLSVmb>J98HcVrF*Y|C#=2#nQ&b)QOl;+{VzwRMgbi 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Donner le résultat arrondi à $10^{-3}$. + \item Interpréter cette valeur dans le contexte de l'exercice. + \end{enumerate} + \end{enumerate} +\end{exercise} +\collectexercisesstop{banque} diff --git a/Tsti2d/Probabilite/Loi_exponentielle/index.rst b/Tsti2d/Probabilite/Loi_exponentielle/index.rst index 0f27f9d..0e88918 100644 --- a/Tsti2d/Probabilite/Loi_exponentielle/index.rst +++ b/Tsti2d/Probabilite/Loi_exponentielle/index.rst @@ -2,7 +2,7 @@ Loi exponentielle pour l'année 2019-2020 en terminale Tsti2d ############################################################ :date: 2020-04-13 -:modified: 2020-04-13 +:modified: 2020-05-08 :authors: Bertrand Benjamin :category: Tsti2d :tags: Probabilité, Variable aléatoire, Exponentielle @@ -45,10 +45,14 @@ Cours: Cours: -.. image:: 3E_proba_esp.pdf +.. image:: 3B_esperance.pdf :height: 200px :alt: Formule de l'espérance Étape 4: Exercices type bac ============================ + +.. image:: 4E_annales.pdf + :height: 200px + :alt: Exercices d'annales sur la loi exponentielle diff --git a/Tsti2d/semaines.rst b/Tsti2d/semaines.rst index 9992730..12c0eea 100644 --- a/Tsti2d/semaines.rst +++ b/Tsti2d/semaines.rst @@ -2,7 +2,7 @@ Programmes semaine par semaine pendant le confinement avec les Tsti2d ##################################################################### :date: 2020-04-14 -:modified: 2020-05-07 +:modified: 2020-05-08 :authors: Bertrand Benjamin :category: Tsti2d :tags: Progression @@ -51,7 +51,13 @@ S20 - Questions flashs (3x5min) S20 - Annales loi exponentielles (30min) ---------------------------------------- -**Exercice à rendre** +Deux exercices complets du bac de l'année dernière. Seules les parties B sont sur la loi exponentielle. Je vous devez à les faire en entier (sauf la partie C de l'exercice 1 qui est sur un chapitre que l'on a pas encore fait). + +.. image:: 4E_annales.pdf + :height: 200px + :alt: Exercices d'annales sur la loi exponentielle + +**À rendre: exercice 1 partie A et B OU exercice 2 partie B** S20 - Opérations et limites ---------------------------