grandes-ecoles 2018 Q11

grandes-ecoles · France · centrale-maths1__official Inequalities Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.)

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative real $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Show that, for all real $t$,
$$-b(t - |\delta|)^{2} \leqslant a - \frac{1}{2}bt^{2}$$

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative real $t$,

$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$

Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Show that, for all real $t$,

$$-b(t - |\delta|)^{2} \leqslant a - \frac{1}{2}bt^{2}$$

Paper Questions

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