grandes-ecoles 2018 Q12

grandes-ecoles · France · centrale-maths1__official Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof

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 $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Deduce that for all real $t$ such that $t \geqslant |\delta|$ we have
$$\mathbb{P}(|X + \delta| \geqslant t) \leqslant a \exp(a) \exp\left(-\frac{1}{2}bt^{2}\right)$$

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 $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Deduce that for all real $t$ such that $t \geqslant |\delta|$ we have

$$\mathbb{P}(|X + \delta| \geqslant t) \leqslant a \exp(a) \exp\left(-\frac{1}{2}bt^{2}\right)$$

Paper Questions

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