Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is finite and we use the notation from question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that $\phi _ { X }$ is expandable as a power series on $\mathbb { R }$ and, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( \mathrm { i } t ) ^ { n } } { n ! } \mathbb { E } \left( X ^ { n } \right)$.
Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is finite and we use the notation from question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$.\\
Show that $\phi _ { X }$ is expandable as a power series on $\mathbb { R }$ and, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( \mathrm { i } t ) ^ { n } } { n ! } \mathbb { E } \left( X ^ { n } \right)$.