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 countable and we use the notation from question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We also assume that, for all integer $n \in \mathbb { N } , X$ admits a moment of order $n$ and that there exists a real $R > 0$ such that $$\mathbb { E } \left( | X | ^ { n } \right) = O \left( \frac { n ^ { n } } { R ^ { n } } \right) \quad \text { when } n \rightarrow + \infty$$ Using the result of Q39, deduce that for all real $t \in \left[ - \frac { R } { \mathrm { e } } , \frac { R } { \mathrm { e } } \right]$, $$\phi _ { X } ( t ) = \sum _ { k = 0 } ^ { + \infty } \frac { ( \mathrm { i } t ) ^ { k } } { k ! } \mathbb { E } \left( X ^ { k } \right)$$