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zV=pa_wqCP4_`{EtyZt|ytljPY;m-Q%3%4Ba{%yK$dRO_{zYDf1KkTmAs{Qc1Xsh*O l>%V4?N6a7iv$&o9fMay}Wl=_P9y1FAV*@T#RabvEE&xG*;IRMz diff --git a/TDs/M61B_2024-25_TD7.tex b/TDs/M61B_2024-25_TD7.tex index fc1281a..b53c8f4 100644 --- a/TDs/M61B_2024-25_TD7.tex +++ b/TDs/M61B_2024-25_TD7.tex @@ -121,10 +121,7 @@ % ----------------------------------------------- \begin{exo}\emph{(Loi de Laplace)} - On considère une variable aléatoire $X$ dont la densité est donnée par - \[ - f(x)=ce^{-|x|}. - \] + On considère une variable aléatoire $X$ dont la densité est donnée par $f(x)=ce^{-|x|}$. \begin{enumerate} \item Calculer $c$. \item Démontrer que $X$ admet des moments de tout ordre. Les calculer. @@ -167,6 +164,20 @@ \end{exo} +% ----------------------------------------------- 2 +\begin{exo}\emph{(Loi log-normale, DS2 de 2024)} + + Soient $\mu \in \R$ et $\sigma \in \R^{*}$. Une variable aléatoire $X$ à valeurs dans $]0,+\infty[$ est dite suivre une loi \emph{log-normale} de paramètres $(\mu, \sigma^2)$ si $Y = \log X$ suit la loi gaussienne $\Nor{\mu,\sigma^2}$. On note $\LogNor{\mu,\sigma^2}$ la loi log-normale de paramètres $(\mu, \sigma^2)$. + \begin{enumerate} + \item Exprimer $X$ à l'aide d'une variable $Z$ (à préciser) qui suit une loi normale centrée réduite. + En déduire la fonction de répartition de $X$ à l'aide de la fonction $\Phi$ de répartition de $Z$. + Calculer ensuite explicitement la densité de $X$. + \item Calculer l'espérance de $X$. + \item Montrer que $X^r\sim\LogNor{r\mu,r^2\sigma^2}$ pour tout $r \neq 0$. En déduire la valeur de $\EE{X^r}$ pour tout $r\in\R$. Calculer la variance de $X$. + \item Calculer $\EE{e^{uX}}$ pour tout $u >0$. + \end{enumerate} +\end{exo} + % ----------------------------------------------- \begin{exo}\emph{(Une loi à reconnaître)} -- GitLab