During a laying, an insect lays a random number of eggs following a Poisson distribution with parameter $\lambda>0$. Then, the probability that a given egg becomes a new insect is $\alpha\in]0,1[$.
Recall the generating function of a random variable following a Poisson distribution with parameter $\lambda$.

During a laying, an insect lays a random number of eggs following a Poisson distribution with parameter $\lambda>0$. Then, the probability that a given egg becomes a new insect is $\alpha\in]0,1[$.
Using the composition relation $G_S=G_T\circ G_X$, determine the distribution of the number of insects resulting from the laying.

Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$
Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$. For all integers $n \geqslant 1$, determine the distribution of $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$.

We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set
$$\forall n \in \mathbb{N}^{*}, \quad S_{n} = X_{1} + \cdots + X_{n}$$
VI.A.1) By induction, prove that, for every integer $n \in \mathbb{N}$, $S_{n}$ follows the Poisson distribution with parameter $n\lambda$.
We will admit that, for every integer $n \in \mathbb{N}^{*}$, the variables $S_{n}$ and $X_{n+1}$ are mutually independent.
VI.A.2) Let $\varepsilon \in \mathbb{R}_{+}^{*}$. Prove that
$$\forall n \in \mathbb{N}^{*}, \quad P\left(\left|S_{n} - n\lambda\right| \geqslant n\varepsilon\right) \leqslant \frac{\lambda}{n\varepsilon^{2}}$$
VI.A.3) Let $\varepsilon > 0$. Justify the following two inclusions
$$\begin{aligned} & \left(S_{n} > n(\lambda+\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \\ & \left(S_{n} \leqslant n(\lambda-\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \end{aligned}$$
VI.A.4) In all the following questions, we assume $x \geqslant 0$.
Deduce from VI.A.3 that
$$\begin{cases} \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 0 & \text{if } 0 \leqslant x < \lambda \\ \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 1 & \text{if } x > \lambda \end{cases}$$

We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set $S_{n} = X_{1} + \cdots + X_{n}$.
Using question VI.A, show that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \begin{cases} 0 & \text{if } 0 \leqslant x < \lambda \\ 1 & \text{if } x > \lambda \end{cases}$$

Justify that for all $\ell \geqslant 0$ and $n \in \mathbb{N}$, $(N(0,\ell) = n+1) = (S_n \leqslant \ell < S_{n+1})$ up to a negligible set. Deduce that, up to negligible sets, $$\left(S_n \leqslant \ell\right) = (N(0,\ell) \geqslant n+1) \quad \text{and} \quad \left(S_n \geqslant \ell\right) \subset (N(0,\ell) \leqslant n+1).$$

Suppose in this question that $X$ additionally admits a finite variance $V$. Show then that $$\forall \varepsilon > 0, \forall n \geqslant 1, \quad \mathbb{P}\left(S_n \leqslant n(m-\varepsilon)\right) \leqslant \frac{V}{\varepsilon^2 n}.$$

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We set for every integer $n \geqslant 0$, $B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$ where $S(n,k)$ is the number of partitions of $\llbracket 1,n \rrbracket$ into $k$ parts. Let $m$ be a strictly positive integer, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
We assume in this question that $Y$ follows the Poisson distribution with parameter 1.
IV.C.1) Show that for all $n \in \mathbb { N } , B _ { n } = \mathbb { E } \left( Y ^ { n } \right)$.
IV.C.2) Deduce that for every polynomial $Q ( X )$ with integer coefficients, the series $\sum _ { n = 0 } ^ { + \infty } \frac { Q ( n ) } { n ! }$ is convergent and its sum is of the form $N e$, where $N$ is an integer.

Let $n$ be a non-zero natural integer and let $X_{1}, \ldots, X_{n}$ be mutually independent random variables following Poisson distributions with respective parameters $\lambda_{1}, \ldots, \lambda_{n}$. Show that $X_{1} + \cdots + X_{n}$ follows a Poisson distribution with parameter $\lambda_{1} + \cdots + \lambda_{n}$.

Let $X$ be a Poisson random variable. Show that $X$ is infinitely divisible.

Let $r$ be a non-zero natural integer and let $X_{1}, \ldots, X_{r}$ be mutually independent Poisson random variables. Show that $\sum_{i=1}^{r} i X_{i}$ is an infinitely divisible random variable.

$\lambda$ 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 $\lambda$.
Show that $Z$ admits moments of all orders.

$\lambda$ 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 $m _ { 2 } ( Z )$.

Show that the random variable $S_n$ follows a Poisson distribution and specify its parameter.

Verify that, for all $n \in \mathbb{N}^*$, $$n! \left(\frac{2}{n}\right)^n P\left(S_n > n\right) = \mathrm{e}^{-n/2} \sum_{k=1}^{\infty} \frac{n! n^k}{(n+k)!} \left(\frac{1}{2}\right)^k.$$

Let $n \in \mathbb{N}^*$. Show that for all $k \in \mathbb{N}^*$, $$\left(\frac{n}{n+k}\right)^k \leqslant \frac{n! n^k}{(n+k)!} \leqslant 1.$$

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day. Prove that $X$ follows a Poisson distribution with parameter $\lambda p$. Give the expectation and variance of $X$.

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ Using Markov's inequality, prove that if $p \leqslant 2 \frac { 1 - \alpha } { \lambda }$, then condition (II.1) is satisfied.

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ We set $x = -(\lambda p + 1)$. Prove that condition (II.1) is equivalent to the condition $$x \mathrm { e } ^ { x } \leqslant - \alpha \mathrm { e } ^ { - 1 }.$$

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, discuss according to the position of $\lambda$ with respect to $-1 - V\left(-\alpha \mathrm{e}^{-1}\right)$ the existence of a largest real $p \in ]0,1[$ satisfying condition (II.1).

In this question, we fix a natural number $k$. Determine $$y_k = \lim_{n \rightarrow +\infty} P_n\left(X_n = k\right).$$ Let $Y$ be a random variable on a probability space $(\Omega, \mathcal{A}, P)$, taking values in $\mathbf{N}$, and satisfying $$\forall k \in \mathbf{N} \quad P(Y = k) = y_k.$$ Identify the distribution of $Y$.

Let $X$ be a Bernoulli random variable with parameter $\lambda \in ]0,1[$. Show that $$d_{VT}\left(p_X, \pi_\lambda\right) = \lambda\left(1 - e^{-\lambda}\right).$$ Deduce that $$d_{VT}\left(p_X, \pi_\lambda\right) \leq \lambda^2.$$

Verify the relation, for all non-zero natural number $n$, $$2 d_{VT}\left(p_{X_n}, \pi_1\right) = \sum_{k=0}^{n} \frac{1}{k!} \left|\sum_{i=n-k+1}^{+\infty} \frac{(-1)^i}{i!}\right| + e^{-1} \sum_{k=n+1}^{+\infty} \frac{1}{k!}.$$

By continuing to bound the right-hand side of the equality in question 13, establish the estimate $$d_{VT}\left(p_{X_n}, \pi_1\right) \underset{n \rightarrow +\infty}{=} O\left(\frac{2^n}{(n+1)!}\right)$$ One may use binomial coefficients.

Let $\alpha$ and $\beta$ be two strictly positive real numbers. Using the results and methods above, show that $$d_{VT}\left(\pi_\alpha, \pi_\beta\right) \leq |\beta - \alpha|.$$