We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Let $j$ be an integer such that $1 \leqslant j \leqslant k$. Show that for all real $x , | x | ^ { j } \leqslant 1 + | x | ^ { k }$ and deduce that $X$ admits a moment of order $j$.
We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$.\\
Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$.\\
Let $j$ be an integer such that $1 \leqslant j \leqslant k$. Show that for all real $x , | x | ^ { j } \leqslant 1 + | x | ^ { k }$ and deduce that $X$ admits a moment of order $j$.