a) Suppose that $X$ is bounded. Justify that $X$ satisfies $(C_{\tau})$ for all $\tau$ in $\mathbb{R}^{+*}$.
b) Suppose that $X$ follows the geometric distribution with parameter $p \in ]0,1[$ $$\forall k \in \mathbb{N}^{*}, \quad P(X = k) = p(1-p)^{k-1}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.
c) Suppose that $X$ follows the Poisson distribution with parameter $\lambda$: $$\forall k \in \mathbb{N}, \quad P(X = k) = \mathrm{e}^{-\lambda} \frac{\lambda^{k}}{k!} \quad \text{where } \lambda \in \mathbb{R}^{+*}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.

Let $a$ and $b$ be two real numbers such that $a < b$. Suppose $E\left(\mathrm{e}^{aX}\right) < +\infty$ and $E\left(\mathrm{e}^{bX}\right) < +\infty$.
a) Show $\forall t \in [a,b]$, $\mathrm{e}^{tX} \leqslant \mathrm{e}^{aX} + \mathrm{e}^{bX}$. Deduce that $E\left(\mathrm{e}^{tX}\right) < +\infty$. What can we conclude about the set $\left\{t \in \mathbb{R} ; E\left(\mathrm{e}^{tX}\right) < +\infty\right\}$?
b) Let $k$ be in $\mathbb{N}$, $t$ in $]a,b[$. We denote by $\theta_{k,t,a,b}$ the function $y \in \mathbb{R} \mapsto \frac{y^{k} \mathrm{e}^{ty}}{\mathrm{e}^{ay} + \mathrm{e}^{by}}$. Determine the limits of $\theta_{k,t,a,b}$ at $+\infty$ and $-\infty$. Show that this function is bounded on $\mathbb{R}$.
c) Show that $E\left(|X|^{k} \mathrm{e}^{tX}\right) < +\infty$.
d) We return to the notations of question b). Let $k$ be in $\mathbb{N}$, $c$ and $d$ be two real numbers such that $a < c < d < b$. Show that there exists $M_{k,a,b,c,d} \in \mathbb{R}^{+}$ such that for all $t \in [c,d]$ and for all $y \in \mathbb{R}$: $\left|\theta_{k,t,a,b}(y)\right| \leqslant M_{k,a,b,c,d}$.

In this question, $\tau$ is an element of $\mathbb{R}^{+*}$ and $X$ satisfies $(C_{\tau})$.
a) Show that the set of real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$ is an interval $I$ containing $[-\tau, \tau]$. For $t$ in $I$, we denote $\varphi_{X}(t) = E\left(\mathrm{e}^{tX}\right)$.
b) Show that if $X(\Omega)$ is finite, $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
c) Suppose now that $X(\Omega)$ is a countably infinite set. We denote $X(\Omega) = \left\{x_{n} ; n \in \mathbb{N}^{*}\right\}$ where $\left(x_{n}\right)_{n \in \mathbb{N}^{*}}$ is a sequence of pairwise distinct real numbers and we set for all $n \in \mathbb{N}^{*}$, $p_{n} = P\left(X = x_{n}\right)$. Using the results established in question I.A.3 and two theorems relating to series of functions which you will state completely, show that $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
d) Verify that for $t$ in the interior of $I$ and $k$ in $\mathbb{N}$, $\varphi_{X}^{(k)}(t) = E\left(X^{k} \mathrm{e}^{tX}\right)$.
e) Let $\psi_{X} = \frac{\varphi_{X}^{\prime}}{\varphi_{X}}$. Show that $\psi_{X}$ is increasing on $I$ and that, if $X$ is not almost surely equal to a constant, $\psi_{X}$ is strictly increasing on $I$.

Let $\alpha$ be a strictly positive real and $X$ a discrete random variable admitting an exponential moment of order $\alpha$. Show that the random variable $e^{\alpha X}$ has finite expectation.

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.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
Show that the random variable $X$ admits an exponential moment of order $\alpha$ for every strictly positive real number $\alpha$.

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Using the results from the preamble, show that, for all $t \in ] - R _ { X } , R _ { X } [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation and that $M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right)$.

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Show conversely that, if there exists a real $R > 0$ such that, for all $t \in ] - R , R [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation, then the domain of definition of the moment generating function of $X$ contains $] - R , R [$ and for all $t \in ] - R , R \left[ , M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right) \right.$.

Show that for all $t \in \mathbb { R } , \left| \phi _ { X } ( t ) \right| \leqslant 1$.

Let $X$ be a real random variable. Show that $\left| \Phi _ { X } ( \theta ) \right| \leq 1$ for all real $\theta$.