Chapitres Maths en ECG1
Chapitres Maths en ECG1
Corrigés : Variables aléatoires à densité en ECG1
Résumé de cours Exercices Corrigés
Cours en ligne de Maths en ECG1
Corrigés – Variables aléatoires à densité
Exercice :
1) a) Soit par définition de
on a
![Rendered by QuickLaTeX.com F_Y \left( x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-6a7445610d17d278adf922ab7e322668_l3.png)
![Rendered by QuickLaTeX.com = \mathbb{P} \left( Y \le x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8f04511d94ff476fdf7410d435d4c0ca_l3.png)
![Rendered by QuickLaTeX.com = \mathbb{P} \left( - x \le Y \le x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bdb320f99b61b97bc3aedda0f300592e_l3.png)
![Rendered by QuickLaTeX.com = \phi \left( x \right) - \phi \left( - x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8368c6be6474e5b639cb4e25f2f517da_l3.png)
![Rendered by QuickLaTeX.com \phi \left( - x \right) = 1 - \phi \left( x \right),](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-415768a66f18c15513d58778743d0e80_l3.png)
b) Comme prend des valeurs positives, on a
si
On peut résumer la fonction de répartition de
de la fa\c{c}on suivante :
La fonction est dérivable sur
sauf peut-être en
Ainsi
admet une densité
donnée par
soit
c) Pour montrer que admet une espérance, montrons que
converge (convergence absolue).
![Rendered by QuickLaTeX.com f_Y](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d0ad17f2a834e08eaf191b89a8c42d11_l3.png)
![Rendered by QuickLaTeX.com \left] - \infty, 0 \right],](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d5e863b3d360b266cfdf1bdf9eee9195_l3.png)
![Rendered by QuickLaTeX.com \int_0^{+ \infty} t f_Y \left( t \right) dt.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ffb1463a7da063aea46de10e38e42af5_l3.png)
![Rendered by QuickLaTeX.com \lim_{t \to + \infty} t^2 \left( t \sqrt{\dfrac{2}{\pi}} e^{- \dfrac{-t^2}{2}} \right) = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8d1950d7a4ff8b304407023f6bf5a31e_l3.png)
![Rendered by QuickLaTeX.com t f_Y \left( t \right) \underset{t \to + \infty}{=} o \left( \dfrac{1}{t^2} \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a361cb7f1f1713d589c44a7ce122e4c2_l3.png)
Comme l’intégrale
![Rendered by QuickLaTeX.com \int_1^{+ \infty} \dfrac{1}{t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5b7a914098870b15111cb2c0e5a3eb24_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} t f_Y \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-fbab1725fd18a9058eca0cd57b0a78af_l3.png)
![Rendered by QuickLaTeX.com \int_0^{+ \infty} t f_Y \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-be072c96cabbaa0466625edcc91ac767_l3.png)
![Rendered by QuickLaTeX.com Y](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png)
![Rendered by QuickLaTeX.com \mathbb{E} \left( Y \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-77245a60a7c5a9b589e6d9bec16197ab_l3.png)
![Rendered by QuickLaTeX.com = \sqrt{\dfrac{2}{\pi}} \int_0^{+ \infty} t e^{- \dfrac{t^2}{2}} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-42b054a6055896c1ab8a2b894e86a65f_l3.png)
Soit
![Rendered by QuickLaTeX.com A \ge 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-70615eb90e6f84af3c5646c8b07cb9a4_l3.png)
![Rendered by QuickLaTeX.com \sqrt{\dfrac{2}{\pi}} \int_0^{A} t e^{- \dfrac{t^2}{2}}dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e8d9528e65b459e93a1e2e9bc6b505b7_l3.png)
![Rendered by QuickLaTeX.com = - \sqrt{\dfrac{2}{\pi}} \int_0^{A } \left( - t \right) e^{- \dfrac{t^2}{2}} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-8ad865382ba90255c2864d5633ab04bf_l3.png)
![Rendered by QuickLaTeX.com = - \sqrt{\dfrac{2}{\pi}} \left[ e^{- \dfrac{t^2}{2}} \right]_0^A](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0a64606586813a8fd2655bc2738d302e_l3.png)
![Rendered by QuickLaTeX.com = - \sqrt{\dfrac{2}{\pi}} \left( e^{- \dfrac{A^2}{2}} - 1 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cd07be91ad35358fc219f45565c1f93d_l3.png)
Comme on a
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d) Montrons que admet un moment d’ordre
c’est-à-dire montrons que l’intégrale
converge absolument.
![Rendered by QuickLaTeX.com f_Y](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d0ad17f2a834e08eaf191b89a8c42d11_l3.png)
![Rendered by QuickLaTeX.com \left] - \infty , 0 \right],](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9786b88289f2f1362dc034be52200817_l3.png)
![Rendered by QuickLaTeX.com \int_{0}^{+ \infty} x^2 f_Y \left( x \right) dx.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f6aae8a37ca06b29ee98da73440f2d96_l3.png)
![Rendered by QuickLaTeX.com x \mapsto x^2 f_Y \left( x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-22a70c755303bcecb2e51edb828eda18_l3.png)
![Rendered by QuickLaTeX.com \left[ 0 , + \infty \right[](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-57208e8c848a5fffb79d0e65aee5490d_l3.png)
![Rendered by QuickLaTeX.com \lim_{x \to +\infty} x^2 \left( x^2 e^{- \dfrac{x^2}{2}} \right ) = 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d2cd6d7b5c5d095870790dd9248db13a_l3.png)
![Rendered by QuickLaTeX.com x^2 f_Y \left( x \right) \underset{x \to + \infty}{=} o \left( \dfrac{1}{x^2 } \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-07b6f59f48a23209ec3cdb534dad7df3_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} \dfrac{1}{x^2}dx,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4a6cdcbdd58ff98bd54f2b3672215732_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} x^2 f_Y \left( x \right) dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f910b0de80a851f0a9caf1fbd035efb2_l3.png)
![Rendered by QuickLaTeX.com \int_0^{+ \infty} x^2 f_Y \left( x \right) dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-6c533c9f2f8626df8487170ec404169a_l3.png)
![Rendered by QuickLaTeX.com Y](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png)
![Rendered by QuickLaTeX.com 2,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1189766c43abcab17f8edd0316219726_l3.png)
![Rendered by QuickLaTeX.com A \ge 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-70615eb90e6f84af3c5646c8b07cb9a4_l3.png)
![Rendered by QuickLaTeX.com \sqrt{\dfrac{2}{\pi}} \int_0^A x^2 e^{- \dfrac{x^2}{2}} dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-24c255eb08d8d081c90bb3a403f0451a_l3.png)
![Rendered by QuickLaTeX.com u :x \mapsto x](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-164b72d8724ea42d49651eb7f5a498ef_l3.png)
![Rendered by QuickLaTeX.com v : x \mapsto - e^{- \dfrac{x^2}{2}}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-506a64cfd22d66a07f7824fa709cfedf_l3.png)
![Rendered by QuickLaTeX.com u](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png)
![Rendered by QuickLaTeX.com v](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png)
![Rendered by QuickLaTeX.com C^1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b1cd5af4f12e712d1a939f8991cb3e04_l3.png)
![Rendered by QuickLaTeX.com \left[ 0 , A \right]](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-6acdee768f3375cd7aceb126945ac119_l3.png)
![Rendered by QuickLaTeX.com \sqrt{\dfrac{2}{\pi}} \int_0^A x^2 e^{- \dfrac{x^2}{2}} dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-24c255eb08d8d081c90bb3a403f0451a_l3.png)
![Rendered by QuickLaTeX.com = \sqrt{\dfrac{2}{\pi}} ( \left[ -x e^{- \dfrac{x^2}{2}} \right]_0^A](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-605b7329fde4437bcf05a24c368b5b06_l3.png)
![Rendered by QuickLaTeX.com + \int_0^A e^{- \dfrac{x^2}{2}} dx )](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-75e856d6869211c4d3b5d4317647d205_l3.png)
![Rendered by QuickLaTeX.com = -\sqrt{\dfrac{2}{\pi}} A e^{- \dfrac{A^2}{2}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5832bc5922bb256f7e07d0b030be9b1d_l3.png)
![Rendered by QuickLaTeX.com + \sqrt{\dfrac{2}{\pi}} \int_0^A e^{- \dfrac{x^2}{2}} dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b4bcea1b91d2c9de204fff360595d3fe_l3.png)
![Rendered by QuickLaTeX.com = -\sqrt{\dfrac{2}{\pi}} A e^{- \dfrac{A^2}{2}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5832bc5922bb256f7e07d0b030be9b1d_l3.png)
![Rendered by QuickLaTeX.com + 2 \times \dfrac{1}{\sqrt{2 \pi}} \int_0^A e^{- \dfrac{x^2}{2}} dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9b8aeeb359ab425a6f559775915e3772_l3.png)
![Rendered by QuickLaTeX.com \lim_{A \to + \infty} -\sqrt{\dfrac{2}{\pi}} A e^{- \dfrac{A^2}{2}} = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f81b4574b0dcb221c4b45f277384b056_l3.png)
![Rendered by QuickLaTeX.com \lim_{A \to + \infty} \dfrac{1}{\sqrt{2 \pi}} \int_0^A e^{- \dfrac{x^2}{2}} dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-26644b980a5ffb8e912bd4a7bb1740ae_l3.png)
![Rendered by QuickLaTeX.com = \dfrac12,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3091849e2d06ea2dbbeb45a0f1d93f0d_l3.png)
Finalement
2) a) Avant de procéder au changement de variable, vérifions que l’intégrale
converge.
![Rendered by QuickLaTeX.com t \mapsto \dfrac{e^{-t}}{ \sqrt{\pi t} }](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e79bb41d9c13e10e2bc7b80ebe659102_l3.png)
![Rendered by QuickLaTeX.com \left] 0 , + \infty \right[.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4827a6f566e0aeaf0bb3b5c92227031d_l3.png)
![Rendered by QuickLaTeX.com \int_0^1 g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3a2bc44b090873550577823e13fb77b3_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c3ab1efd318f63aecb535932c89706f5_l3.png)
![Rendered by QuickLaTeX.com \star](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ce5ce4f1ed0448af8dbee67daac86254_l3.png)
![Rendered by QuickLaTeX.com \int_0^1 g \left( t \right) dt.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9be76fce859fb5ed0bcc18a5522d6f76_l3.png)
![Rendered by QuickLaTeX.com \dfrac{e^{-t}}{ \sqrt{\pi t} } \underset{t \to 0}{\sim} \dfrac{1}{\sqrt{\pi} \sqrt{t}}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cc07df56757af5d82f023b941bfe6715_l3.png)
![Rendered by QuickLaTeX.com t \mapsto \dfrac{e^{-t}}{ \sqrt{\pi t} }](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e79bb41d9c13e10e2bc7b80ebe659102_l3.png)
![Rendered by QuickLaTeX.com t \mapsto \dfrac{1}{\sqrt{\pi} \sqrt{t}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cb6c4fde0a0ad4a2d20e2ded1a4021c7_l3.png)
![Rendered by QuickLaTeX.com \left] 0 , 1 \right].](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e3d1983537cfcd816ae0c88ac27d6b05_l3.png)
![Rendered by QuickLaTeX.com \int_0^1 \dfrac{1}{\sqrt{\pi} \sqrt{t}} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ab7d01e961b3235f912e61baeddf8999_l3.png)
![Rendered by QuickLaTeX.com \int_0^1 g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3a2bc44b090873550577823e13fb77b3_l3.png)
![Rendered by QuickLaTeX.com \star](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ce5ce4f1ed0448af8dbee67daac86254_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} g \left( t \right) dt.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4bd36949e7444be4ed99f4ed8377f4ea_l3.png)
![Rendered by QuickLaTeX.com \lim_{t \to + \infty} t^2 g \left( t \right) = 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-60e41180af3ae405ad5cb507a1e048bf_l3.png)
![Rendered by QuickLaTeX.com g \left( t \right) \underset{t \to + \infty}{=} o \left( \dfrac{1}{t^2} \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-70d41f872cae2dbeef12c8636566c4c4_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} \dfrac{1}{t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5b7a914098870b15111cb2c0e5a3eb24_l3.png)
![Rendered by QuickLaTeX.com \int_1^{+ \infty} g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c3ab1efd318f63aecb535932c89706f5_l3.png)
![Rendered by QuickLaTeX.com \int_0^{+ \infty} g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a1b4f6907ba2d806b65523cc4aaa308e_l3.png)
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com A \ge 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-56896fd134ec397177b37263272223e0_l3.png)
![Rendered by QuickLaTeX.com \int_0^A g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c2cbf659d7687c2114c3d3cf451c679e_l3.png)
![Rendered by QuickLaTeX.com = \int_0^{\sqrt{2A}} \dfrac{e^{- \dfrac{u^2}{2}}}{\sqrt{\pi \times \dfrac{u^2}{2} }} u du](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-cbeaa72fecb8982ae1cb5383b01f3684_l3.png)
![Rendered by QuickLaTeX.com = \sqrt{\dfrac{2}{\pi}} \int_0^{A} e^{- \dfrac{u^2}{2}} du](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-84a4fd68ab9e60eb935330a83b6bb5fd_l3.png)
![Rendered by QuickLaTeX.com A](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png)
![Rendered by QuickLaTeX.com + \infty,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5db06aac704a7a699035ba8b6100f070_l3.png)
![Rendered by QuickLaTeX.com \int_0^{+ \infty} g \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a1b4f6907ba2d806b65523cc4aaa308e_l3.png)
![Rendered by QuickLaTeX.com = \sqrt{\dfrac{2}{\pi}} \int_0^{+ \infty} e^{- \dfrac{u^2}{2}} du.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3e12c2f8af0e020c9d722d9853bfa250_l3.png)
b)
est à valeurs positives et continue sauf en
![Rendered by QuickLaTeX.com \bullet](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2b6e225d778ccd32cb2bd9cc4eaead9a_l3.png)
![Rendered by QuickLaTeX.com \int_{- \infty}^{+ \infty} g \left( t \right) dt = 1.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bf7e6fa71d218e8c59bf72ce06dc6db0_l3.png)
![Rendered by QuickLaTeX.com \dfrac{1}{\sqrt{2 \pi}} \int_{- \infty}^{+ \infty} e^{- \dfrac{u^2}{2}} du =1,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-77b8e266b6e745d5cda3a398ef446f91_l3.png)
![Rendered by QuickLaTeX.com u \mapsto e^{- \dfrac{u^2}{2}},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9ca6471133315855bc89718a04ab1a19_l3.png)
![Rendered by QuickLaTeX.com \int_{- \infty}^{+ \infty} e^{- \dfrac{u^2}{2}} du](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2c57143d3eec97792fb19c2ea8688f70_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{\sqrt{2 \pi}}{2}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f6896a1ca2199f3c62a666632e698077_l3.png)
![Rendered by QuickLaTeX.com = \sqrt{\dfrac{\pi}{2}}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-dafcfd085b111c1fa6a0e57aa372300f_l3.png)
3) a) Soit
![Rendered by QuickLaTeX.com \star](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ce5ce4f1ed0448af8dbee67daac86254_l3.png)
![Rendered by QuickLaTeX.com x < 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4026188a6d5fba596c7bd1bde701b618_l3.png)
![Rendered by QuickLaTeX.com F_T \left( x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a84a0fc33ae658c26b85a44f85cf70df_l3.png)
![Rendered by QuickLaTeX.com = \mathbb{P} \left( \sqrt{2Z} \le x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c59cc70de943c23e8872b60b108f7fd2_l3.png)
![Rendered by QuickLaTeX.com = 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-78875bc33735907a10d2ce7a75c624e8_l3.png)
![Rendered by QuickLaTeX.com = G \left( x \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-3794f2f3a811949f4f9a5a5356c70f00_l3.png)
car
![Rendered by QuickLaTeX.com g](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png)
![Rendered by QuickLaTeX.com \left] - \infty , 0 \right].](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e4533496d31d0b55ef5a62b06fd42c2f_l3.png)
On suppose
On a
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