In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. a) Show that the function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{t X}\right)$ is defined and continuous on the segment $[-\alpha, \alpha]$. b) Show that the function $\Psi$ is differentiable on the interval $]-\alpha, \alpha[$ and determine its derivative function.
In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real.
a) Show that the function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{t X}\right)$ is defined and continuous on the segment $[-\alpha, \alpha]$.
b) Show that the function $\Psi$ is differentiable on the interval $]-\alpha, \alpha[$ and determine its derivative function.