grandes-ecoles 2022 Q18

grandes-ecoles · France · mines-ponts-maths1__mp Geometric Distribution

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a real $K > 0$ independent of $p$ such that
$$\mathbf { E } \left( ( X - \mathbf { E } ( X ) ) ^ { 4 } \right) \leq \frac { K q } { p ^ { 4 } }$$

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.

Deduce that there exists a real $K > 0$ independent of $p$ such that

$$\mathbf { E } \left( ( X - \mathbf { E } ( X ) ) ^ { 4 } \right) \leq \frac { K q } { p ^ { 4 } }$$

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