grandes-ecoles 2022 Q42

grandes-ecoles · France · centrale-maths1__psi Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof

Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$
Deduce that $$\mathbb { P } ( | Z | \geqslant \lambda ) \leqslant 2 \exp \left( - \frac { \lambda ^ { 2 } } { 2 \sigma ^ { 2 } } \right)$$

Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies
$$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$

Deduce that
$$\mathbb { P } ( | Z | \geqslant \lambda ) \leqslant 2 \exp \left( - \frac { \lambda ^ { 2 } } { 2 \sigma ^ { 2 } } \right)$$

Paper Questions

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