grandes-ecoles 2017 QII.D.2

grandes-ecoles · France · centrale-maths2__pc First order differential equations (integrating factor)
In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
We consider $Y$ the real random variable defined by $Y=\frac{1}{2}-\frac{X}{2c}$.
a) Verify that $X=-cY+(1-Y)c$.
b) Show that $\mathrm{e}^{X} \leqslant Y \mathrm{e}^{-c}+(1-Y) \mathrm{e}^{c}$. 
In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.

We consider $Y$ the real random variable defined by $Y=\frac{1}{2}-\frac{X}{2c}$.

a) Verify that $X=-cY+(1-Y)c$.

b) Show that $\mathrm{e}^{X} \leqslant Y \mathrm{e}^{-c}+(1-Y) \mathrm{e}^{c}$.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.D.1 QII.D.2 QII.D.3 QII.D.4 QII.D.5 QII.D.6