We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Express, for $t\in[0,1]$, $f(t)$ and calculate $m$.

Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ We denote $G_{X}(t) = \mathrm{E}\left(t^{X}\right) = \sum_{k=0}^{\infty} \mathrm{P}(X = k) t^{k}$ (generating series of the random variable $X$).
Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Determine $G_{X}(t)$.

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 $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { S _ { n } } ( t )$.

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$, where $g_{n}(t) = \sum_{k=0}^{+\infty} P(X_{n} = k) t^{k}$.
Determine the distributions of $X_{1}, X_{2}$ and $X_{3}$ and then the functions $g_{1}, g_{2}$ and $g_{3}$.

Explicitly calculate the generating function $G_{X_1}$ of the random variable $X_1$, where $X_1$ follows a Poisson distribution with parameter $1/2$.

Let $n \in \mathbb { N } ^ { * }$ and $\left. p \in \right] 0,1 [$. We assume that $X : \Omega \rightarrow \mathbb { R }$ follows a binomial distribution $\mathcal { B } ( n , p )$ and we denote $q = 1 - p$. Show that, for all $t \in \mathbb { R } , \phi _ { X } ( t ) = \left( q + p \mathrm { e } ^ { \mathrm { i } t } \right) ^ { n }$.

Let $\lambda > 0$. What is the characteristic function of a random variable following a Poisson distribution with parameter $\lambda$ ?

We denote by $G_{X_n}$ and $G_Y$ the generating functions of the variables $X_n$ and $Y$ from the previous question, respectively. Express $G_{X_n}(s)$ as a sum, for $s$ real, and verify that $$\forall s \in \mathbf{R} \quad \lim_{n \rightarrow +\infty} G_{X_n}(s) = G_Y(s)$$