For each of the following real random variables, determine the strictly positive reals $\alpha$ such that the random variable admits an exponential moment of order $\alpha$ and calculate $\mathbb{E}\left(\mathrm{e}^{\alpha X}\right)$ in this case.
a) $X$ a random variable following a Poisson distribution with parameter $\lambda$, where $\lambda$ is a strictly positive real.
b) $Y$ a random variable following a geometric distribution with parameter $p$, where $p$ is a real strictly between 0 and 1.
c) $Z$ a random variable following a binomial distribution with parameters $n$ and $p$, where $n$ is a strictly positive integer and $p$ is a real strictly between 0 and 1.

Deduce that, for any real $\xi$, $\int_{-\infty}^{+\infty} \exp\left(-x^{2}\right) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \sqrt{\pi} \exp\left(-\pi^{2} \xi^{2}\right)$.

We set $\sigma^{\prime} = \frac{1}{2\pi\sigma}$. Show that there exists a real $\mu$ such that $\mathcal{F}\left(g_{\sigma}\right) = \mu g_{\sigma^{\prime}}$. The value of $\mu$ need not be made explicit.

$\lambda$ is a fixed real number. We assume that $Z$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the Poisson distribution with parameter $\lambda$.
Calculate the moment generating function of $Z$. Deduce the values of $m _ { 1 } ( Z )$ and $m _ { 2 } ( Z )$.

Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
Calculate the moment generating function of the random variable $S _ { n }$.

For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.
Calculate the moment generating function of the random variable $Y _ { n }$.

We assume in this question that $X ( \Omega )$ is a finite set of cardinality $r \in \mathbb { N } ^ { * }$. We denote $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ where the $x _ { i }$ are pairwise distinct, and, for all integer $k \in \llbracket 1 , r \rrbracket , a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all real $t , \phi _ { X } ( t ) = \sum _ { k = 1 } ^ { r } a _ { k } \mathrm { e } ^ { \mathrm { i } t x _ { k } }$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $( a , b ) \in \mathbf { R } ^ { 2 }$ and all real $\theta$,
$$\Phi _ { a X + b } ( \theta ) = \frac { p e ^ { i ( a + b ) \theta } } { 1 - q e ^ { i a \theta } }$$