grandes-ecoles 2020 Q1

grandes-ecoles · France · x-ens-maths2__mp Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables

Let $Z$ be a discrete real random variable such that $\exp ( \lambda Z )$ has finite expectation for all $\lambda > 0$. Show that for all $\lambda > 0$ and $t \in \mathbb { R }$, $$P [ Z \geqslant t ] \leqslant \exp ( - \lambda t ) E [ \exp ( \lambda Z ) ] .$$

Let $Z$ be a discrete real random variable such that $\exp ( \lambda Z )$ has finite expectation for all $\lambda > 0$. Show that for all $\lambda > 0$ and $t \in \mathbb { R }$,
$$P [ Z \geqslant t ] \leqslant \exp ( - \lambda t ) E [ \exp ( \lambda Z ) ] .$$

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