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\par
{{\bf UNIVERSIT\'E  DE NICE-SOPHIA ANTIPOLIS \par
  L3 MASS - Int\'egration et Probabilit\'es  - 2006-2007 - 1er semestre \par
  Sylvain Rubenthaler \par
  } {\tt http://math1.unice.fr/$\sim$rubentha/cours.html} }\par



\par \vspace{1cm}
\centerline{{\bf Corrig\'e de la feuille d'exercices num\'ero  9}}
\centerline{{\bf }}
\par \vspace{1.5cm}


\begin{enumerate}


\item La fonction $t\in \R \mapsto |t|^{q/p}$ est convexe donc par in\'egalit\'e de Jensen, \\
$\E(|X|^p)^{q/p}\leq \E((|X|^p)^{q/p}) =\E(|X|^q)$.

\item
\begin{eqnarray*}
\sum_{k \in \N, a_n\leq k \leq \inf(b_n,n)} C_n^k p^k(1-p)^{n-k} & = & \p(Y_n \in [a_n,b_n])\\
& = & \p(X_1+\dots+X_n \in [\sqrt{p(1-p)n}a+np,\sqrt{p(1-p)n}b+np])\\
& = & \p(\frac{X_1+\dots+X_n-np}{\sqrt{p(1-p) n}} \in [a,b])\\
\text{TCL} & \convn & \int_a^b \frac{e^{-t^2/2}}{\sqrt{2\pi}}.
\end{eqnarray*}

\item
\begin{eqnarray*}
\p(X>t_1+t_2|X>t_1) & = & \frac{\p(X>t_1+t_2,X>t_1)}{\p(X>t_1)}\\
& = & \frac{\p(X>t_1+t_2)}{\p(X>t_1)}\\
& = & \frac{\int_{t_1+t_2}^{+\infty} \lambda e^{-\lambda t} dt}{\int_{t_1}^{+\infty} \lambda e^{-\lambda t} dt}\\
& = & \frac{e^{-\lambda (t_1+t_2)}}{e^{-\lambda t_1}}\\
& = & e^{-\lambda t_2}\\
& = & \p(X>t_2)\ .
\end{eqnarray*}

\end{enumerate}
\end{document}
