grandes-ecoles 2017 QI.A.1

grandes-ecoles · France · centrale-maths1__psi Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof

Let $U$ and $V$ be two random variables on $(\Omega, \mathcal{A}, P)$ possessing a second moment and such that $V$ is not almost surely zero. Show that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} \geqslant 0$ and that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} = 0$ if and only if there exists $\lambda \in \mathbb{R}$ such that $\lambda V + U$ is almost surely zero.

Let $U$ and $V$ be two random variables on $(\Omega, \mathcal{A}, P)$ possessing a second moment and such that $V$ is not almost surely zero. Show that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} \geqslant 0$ and that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} = 0$ if and only if there exists $\lambda \in \mathbb{R}$ such that $\lambda V + U$ is almost surely zero.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 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