grandes-ecoles 2017 QII.D.3

grandes-ecoles · France · centrale-maths2__pc Moment generating functions Upper bound on MGF (sub-Gaussian or exponential inequalities)
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 $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
a) Show that $\mathbb{E}\left(\mathrm{e}^{X}\right) \leqslant \cosh(c)$.
b) Deduce that $\forall t \in \mathbb{R}^{+*}, \Psi(t) \leqslant \cosh(ct)$. 
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 $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.

a) Show that $\mathbb{E}\left(\mathrm{e}^{X}\right) \leqslant \cosh(c)$.

b) Deduce that $\forall t \in \mathbb{R}^{+*}, \Psi(t) \leqslant \cosh(ct)$.
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