grandes-ecoles 2018 Q42

grandes-ecoles · France · centrale-maths1__official Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables

We consider $g(X)$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable with independent Rademacher coefficients and $g(M) = \|M \cdot u\|$ for a fixed unit vector $u$, and $m$ is a median of $g(X)$. Deduce that $(\sqrt{k} - m)^{2} \leqslant \mathbb{E}\left((g(X) - m)^{2}\right)$.

We consider $g(X)$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable with independent Rademacher coefficients and $g(M) = \|M \cdot u\|$ for a fixed unit vector $u$, and $m$ is a median of $g(X)$. Deduce that $(\sqrt{k} - m)^{2} \leqslant \mathbb{E}\left((g(X) - m)^{2}\right)$.

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