diff --git a/cours.tex b/cours.tex index 7d3aa419e7f1ffbca78afce4f5f810789f957c43..9af1025ce1e1c0ffd94bfa4ee19d55f74776b061 100644 --- a/cours.tex +++ b/cours.tex @@ -2994,10 +2994,308 @@ le moins complexe. Et la plus grande difficulté tient dans le ``représentatif Il existe différentes façon de représenter les caractères d'une population selon que sa nature est \textit{discrète} ou \textit{continue}. Dans le cas discret d'un caractère pouvant prendre $k\in\natural$ valeur différentes $\{x_i\}_{i=0}^{k-1}$, -on représente le nombre d'indivius pouvant prendre la valeur $x_k$ par le nombre $n_k$. On a donc un ensemble $\{n_i\}_{i=0}^{n-1}$ +on représente le nombre d'indivius pouvant prendre la valeur $x_i$ par le nombre $n_i$. On a donc un ensemble $\{n_i\}_{i=0}^{k-1}$ d'indivius pour les $k$ valeurs des caratères de la population. Dans le cas continu le nombre d'individus d'un caractère -correspondrant à une subdivision en $k$ parties de l'ensemble des valeurs possibles pour le dit caractère. On peut résumer les deux -représentation sous la forme de tableaux. Dans le cas discret, on reprendre l'exemple de +correspondrait à une subdivision en $k$ parties de l'ensemble des valeurs possibles pour le dit caractère. +\begin{exemples}\hfill\break + \begin{enumerate} + \item Cas discret: On étudie la distribution de salaires annuels dans une entreprise. + Les salaires possibles sont $40'000$, $50'000$, $60'000$ et $1'000'000$ de CHF. + \begin{itemize} + \item Il y a 35 personnes payées $40'000$ CHF. + \item Il y a 20 personnes payées $50'000$ CHF. + \item Il y a 5 personnes payées $60'000$ CHF. + \item Il y a 1 personne payée $1'000'000$ CHF. + \end{itemize} + + \item Cas continu: Lors du benchmark d'une application, $A$, nous effetuons plusieurs mesures (la population) du temps d'exécution (le caractère) de l'application. + Les résultats obtenus sont les suivants: + \begin{itemize} + \item 7 exécutions ont pris entre 50 et 51 secondes. + \item 12 exécutions ont pris entre 51 et 52 secondes. + \item 8 exécutions ont pris entre 52 et 53 secondes. + \item 23 exécutions ont pris entre 53 et 54 secondes. + \end{itemize} + + \end{enumerate} + +\end{exemples} +Pour représenter de façon un peu plus parlante ces valeurs, deux méthodes principales existent: le tableau ou le graphique. +Pour illustrer les exemples précédents sous forme de tableau on obtient pour le cas des salaires (voir Tabl.~\ref{fig_salaires}) +\begin{table} +\begin{center} +\begin{tabular}{|c|c|} +\hline + Salaire & Nombre de salairés \\ + \hline\hline + 40000 & 35 \\ + \hline + 50000 & 20 \\ + \hline + 60000 & 5 \\ + \hline + 1000000 & 1 \\ + \hline +\end{tabular} +\end{center} +\caption{Tableau du nombre de salariés par salaire.}\label{table_salaires} +\end{table} +et du benchmark de l'application (voir Tabl.~\ref{fig_exec}) +\begin{table} +\begin{center} +\begin{tabular}{|c|c|} +\hline + Temps d'exécution & Nombre \\ + \hline\hline + [50,51) & 7 \\ + \hline + [51,52) & 12 \\ + \hline + [52,53) & 8 \\ + \hline + [53,54) & 23 \\ + \hline +\end{tabular} +\end{center} +\caption{Tableau des temps d'exécution.}\label{table_exec} +\end{table} +Sous forme de graphique on peut représenter le tableau des salaires sous la forme d'un graphique bâton (voir Fig.~\ref{fig_salaires}) +\begin{figure}[htp] +\begin{center} +\includegraphics[width=0.5\textwidth]{figs/graph_salaires.pdf} +\caption{Nombre salariés en fonction du salaire.}\label{fig_salaires} +\end{center} +\end{figure} +ou d'un histogramme pour le temps d'exécution de l'application (voir Fig.~\ref{fig_exec}). +\begin{figure}[htp] +\begin{center} +\includegraphics[width=0.5\textwidth]{figs/graph_exec.pdf} +\caption{Nombre d'exécutions en fonction du temps d'exécution.}\label{fig_exec} +\end{center} +\end{figure} + +\subsection{Fréquences} + +Plutôt que de faire apparaître le nombre d'individus d'une population +possédant un caractère, il peut être plus intéressant et parlant de faire intervenir +la \textit{fréquence} ou le nombre relatif à la place. En effet, la fréquence donne +immédiatement la proportion d'individu plutôt qu'un nombre absolu qui n'est pas forcément +très interprétable tout seul. + +La population totale, $n$, est donnée par +\begin{equation} + n=\sum_{i=0}^{k-1}n_i. +\end{equation} +On peut donc définir la fréquence d'un caractère $i$, $f_i$ comme +\begin{equation} + f_i=\frac{n_i}{n}. +\end{equation} +\begin{exemples}{Fréquence} + +Les tableaux de fréquence des deux exemples précédents sont donnés par + \begin{enumerate} + \item Cas discret: la population totale est de + \begin{equation} +% n=40'000+50'000+60'000+1'000'000=1'150'000. + n=35+20+5+1=61. + \end{equation} + \begin{table}[htp] + \begin{center} + \begin{tabular}{|c|c|c|} + \hline + Salaire & Nombre de salairés & Fréquence\\ + \hline\hline + 40000 & 35 & $35/61\cong0.573770$\\ + \hline + 50000 & 20 & $20/61\cong0.327869$\\ + \hline + 60000 & 5 & $5/61\cong0.081967$ \\ + \hline + 1000000 & 1 & $1/61\cong0.19568$ \\ + \hline + \end{tabular} + \end{center} + \caption{Tableau des salaires, du nombre de salariés et la fréquence.} + \end{table} + + \item Cas continu: la population totale est de + \begin{equation} + n=7+12+8+23=50. + \end{equation} + Le tableau \ref{table_exec_freq} affiche les différentes fréquences des temps d'exécution. + \begin{table}[htp] +\begin{center} +\begin{tabular}{|c|c|c|} +\hline + Temps d'exécution & Nombre & Fréquence \\ + \hline\hline + [50,51) & 7 & $7/50=0.14$\\ + \hline + [51,52) & 12 & $12/50=0.24$ \\ + \hline + [52,53) & 8 & $8/50=0.16$ \\ + \hline + [53,54) & 23 & $23/50=0.46$ \\ + \hline +\end{tabular} +\end{center} +\caption{Tableau des temps d'exécution et la fréquence des temps d'exécution.}\label{table_exec_freq} +\end{table} + + \end{enumerate} +\end{exemples} +La fréquence possède un certain nombre de propriétés que nous retrouverons +dans les sections suivantes qui sont assez intuitives +\begin{proprietes}{Propriétés de la fréquence} + \begin{enumerate} + \item Les fréquences sont toujours dans l'intervalle $[0,1]$ + \begin{equation} + 0\leq f_i\leq 1. + \end{equation} + \item La somme de toutes les fréquences donne toujours $1$ + \begin{equation} + \sum_{i=0}^{k-1} f_i = 1. + \end{equation} + + \end{enumerate} + +\end{proprietes} +Relié avec la propriété $2$ ci-dessus, il peut également être intéressant d'obtenir la +\textit{fréquence cumulée}, notée $F(x)$, d'un caractère qui se définit comme la fréquence des individus +qui présentent une valeur de caractère $x_i\leq x$. Les tableaux ocrrespondants aux tableaux +\ref{table_salaires} et \ref{table_exec} (voir Tabls. \ref{table_salaires_freqcum} et \ref{table_exec_freqcum}) + \begin{table}[htp] + \begin{center} + \begin{tabular}{|c|c|c|c|} + \hline + Salaire & Nombre de salairés & Fréquence & Fréquence cumulée\\ + \hline\hline + 40000 & 35 & $35/61\cong0.573770$ & $35/61\cong0.573770$\\ + \hline + 50000 & 20 & $20/61\cong0.327869$ & $(20+35)/61\cong0.90164$\\ + \hline + 60000 & 5 & $5/61\cong0.081967$ & $(20+35+5+1)/61\cong0.98361$\\ + \hline + 1000000 & 1 & $1/61\cong0.19568$ & $(20+35+5+1)/61=1$\\ + \hline + \end{tabular} + \end{center} + \caption{Tableau des salaires, du nombre de salariés, et la fréquence et fréquence cumulée des salaires.}\label{table_salaires_freqcum} + \end{table} + + \begin{table}[htp] +\begin{center} +\begin{tabular}{|c|c|c|c|} +\hline + Temps d'exécution & Nombre & Fréquence & Fréquence cumulée \\ + \hline\hline + [50,51) & 7 & $7/50=0.14$ & $7/50=0.14$ \\ + \hline + [51,52) & 12 & $12/50=0.24$ & $(7+12)/50=0.38$ \\ + \hline + [52,53) & 8 & $8/50=0.16$ & $(7+12+8)/50=0.54$ \\ + \hline + [53,54) & 23 & $23/50=0.46$ & $(7+12+8+23)/50=1$ \\ + \hline +\end{tabular} +\end{center} +\caption{Tableau des temps d'exécution et la fréquence et fréquences cumulées des temps d'exécution.}\label{table_exec_freqcum} +\end{table} +\begin{exercices}{Fréquence cumulée} +\begin{enumerate} + \item Tracer les graphes de la fréquence cumulée pour les deux exemples que nous avons vus. + \item Que pouvons-nous déduire de la forme de la fonction (croissance, valeur maximale)? +\end{enumerate} + + +\end{exercices} + + +\subsection{Mesures de tendance centrale} + +Jusqu'ici le nombre de valeurs étudiées était limité et il est assez simple d'avoir +une vue d'ensemble de la distribution des valeurs des caractères de notre population. +Il est plus aisé d'utiliser une nombre de valeurs beaucoup plus resreint permettant +de résumer les différents caractères et nous allons en voir deux différents qui nous donne une tendance dite centrale: +la moyenne, la médiane. + +La \textit{moyenne}, notée $\bar{x}$ d'un jeu de données s'obtient par la formule suivante +\begin{equation} + \bar{x}=\frac{1}{n}\sum_{i=0}^{k-1}x_i\cdot n_i. +\end{equation} +La moyenne peut également être calculée via les fréquences +\begin{equation} + \bar{x}=\sum_{i=0}^{k-1}f_i\cdot x_i. +\end{equation} +\begin{exercices}{Propriétés de la moyenne} +\begin{enumerate} + \item Démontrer la relation précédente. + \item Démontrer que la moyenne des écart $x_i-\bar{x}$ est nulle. +\end{enumerate} +\end{exercices} + +\begin{exemple}{Moyenne} + +Pour l'exemple des salaires la moyenne est donnée par +\begin{equation} + \bar{x}_{\textrm{salaire}}=\frac{35\cdot40000+20\cdot50000+5\cdot60000+1\cdot1000000}{61}=60656. +\end{equation} +\end{exemple} +On remarque ici que la moyenne des salaires donne une impression erronnée de la situation car elle est très sensible aux valeurs extrême de la distribution. +En effet, tous les salaires à l'exception d'un sont inférieurs à la moyenne. En effet, si on retire le salaire d'un million de notre ensemble de valeurs, +la moyenne de l'échantillon restant devient +\begin{equation} + \bar{x}_{\textrm{salaire}}=\frac{35\cdot40000+20\cdot50000+5\cdot60000}{60}=45000. +\end{equation} +La différence est de l'ordre de $25\%$ par rapport aux $60'000$ CHF obtenus avec toute la population. +Il est donc nécessaire d'utiliser une autre mesure pour illustrer mieux le salaire caractéristique de notre population. +De façon plus générale la moyenne est peu robuste à des valeurs extrêmes dans l'étude d'échantillon. + +Une mesure qui est plus parlante est la \textit{médiane}, notée $\tilde{x}$. La médiane se définit comme la valeur $\tilde{x}$ qui est telle que +la moitié des individus de la population sont ont un $x_i\leq \tilde{x}$ et le reste est telle que $x_i\geq\tilde{x}$. + +Pour l'exemple des salaires le salaire médian est de $40000 CHF$, ce qui reflète beaucoup mieux la distribution des salaire de notre population. +\begin{exercice}{Moyenne, médiane} + + Calculer la moyenne et la médiane pour l'exemple du temps d'exécution (prendre la borne inférieure des intervalles pour chaque temps + d'exécution\footnote{Il y a 7 temps de 50s, 12 de 51s, 8 de 52s et 23 de 53s.}). +\end{exercice} + + +\subsection{Mesures de dispersion} + +Nous avons vu deux mesures donnant une tendance générale des caractères d'une population. Hors cette valeur ne nous dit absolument rien +sur la manière dont ces caractères sont distribués. Sont-ils proches de la moyenne ou de la médiane? Ou en sont-ils au contraire éloignés? +Nous allons voir deux mesures diffréentes dans cette sous-section: la variance (écart-type), et l'intervalle inter-quantile. + +Nous cherchons d'abord à calculer la moyenne des écarts à la moyenne. +Hors, comme on l'a vu dans la sous-section précédente l'écart à la moyenne $x_i-\bar{x}$ est nul en moyenne. Cette grandeurs ne nous apprend rien. +On peut donc s'intéresser plutôt à la moyenne de l'écart quadratique $(x_i-\bar{x})^2$ qui est une quantité toujours positive et donc la moyenne sera +de cette écart quadratique aura toujours une valeurs qui sera positive ou nulle (elle sera nulle uniquement si +$x_i-\bar{x}=0,\forall i$)\footnote{on pourrait aussi étudier la moyenne de $|x_i-\bar{x}|$, mais cela est moins pratique à étudier théoriquement.}. +On définit donc la \textit{variance}, $v$, comme étant la moyenne des écarts quadratiques +\begin{equation} + v=\frac{1}{n}\sum_{i=0}^{k-1}n_i(x_i-\bar{x})^2. +\end{equation} +Si on considère plutôt la racine carrée de la variance, on obtient \textit{l'écart-type} +\begin{equation} + s=\sqrt{v}. +\end{equation} +\begin{exercices}{Variance, écart-type} + +Démontrer les réations suivantes + \begin{enumerate} + \item On peut également calculer la variance avec la fréquence + \begin{equation} + v=\sum_{i=0}^{k-1}f_i(x_i-\bar{x})^2. + \end{equation} + \item On peut également calculer la variance à l'aide de la formule suivante + \begin{equation} + v=\frac{1}{n}\sum_{i=0}^{k-1}n_ix_i^2-\bar{x}^2. + \end{equation} + \end{enumerate} + +\end{exercices} @@ -3009,7 +3307,7 @@ Les nombres aléatoires, bien que pas directement reliés aux probabilités, son qui vont de la cryptographie aux simulations physiques. 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