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 $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Calculate the expectation $\mathrm{E}(X)$, the variance $V(X)$ and the standard deviation of $X$.

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 $\mathcal{P}(\lambda)$, and $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$. Determine the expectation and standard deviation of the random variables $S_{n}$ and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.

We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.
Compare $\mathrm{P}\left(a \leqslant T_{n} \leqslant b\right)$ and $\sum_{k \in I_{n}} \mathrm{P}\left(S_{n} = k\right)$.

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
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \begin{cases} 0 & \text{if } 0 \leqslant x < \lambda \\ 1 & \text{if } x > \lambda \end{cases}$$

$\lambda$ 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.

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 $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
Compare with the results of question 8.

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

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

Deduce that, when $n$ tends to $+\infty$, $$P\left(S_n > n\right) \sim \frac{\mathrm{e}^{-n/2}}{n!} \left(\frac{n}{2}\right)^n.$$

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).

We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Give a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.

We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$. Let $t \in \mathbf{R}_+$. We assume that the number of impulses after time $t$ is given by a random variable $Y_t$ following a Poisson distribution with parameter $t$. For all $j \in \llbracket 1;N \rrbracket$ we denote by $A_{t,j}$ the event ``the system is in state $j$ after time $t$''. Justify that $P(A_{t,j}) = H_t[1,j]$.

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 $Y$.

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

Problem 6
Company A owns multiple factories $i ( i = 1,2 , \cdots )$. Suppose that the probability of producing defective goods in a factory $i$ is $P _ { i }$, and that $N _ { i }$ goods are randomly sampled and shipped from the factory. Here, $P _ { i }$ is sufficiently small, and each factory does not affect any other.
I. Show the probability $f ( i , k )$, which is the probability of $k$ defective goods existing within $N _ { i }$ goods shipped from a factory $i$. Here, $k$ is a non-negative integer.
II. Show that $f ( i , k ) \rightarrow \frac { e ^ { - \lambda _ { i } } \lambda _ { i } ^ { k } } { k ! }$ when $N _ { i } \rightarrow \infty$. Here, when calculating the limit of $f ( i , k ) , \lambda _ { i }$ is a constant, where $\lambda _ { i } = N _ { i } P _ { i }$.
In the following questions, assume that $f ( i , k ) = \frac { e ^ { - \lambda _ { i } } \lambda _ { i } ^ { k } } { k ! }$.
III. Suppose that goods are shipped from two factories as shown in Table 1. Find the probability of two defective goods being contained within all shipped goods.

\begin{tabular}{ c } Factory number

$( i )$

&

Probability of defectiveness

$\left( P _ { i } \right)$

&

Number of shipped goods

$\left( N _ { i } \right)$

\hline 1 & 0.01 & 500 \hline 2 & 0.02 & 300 \hline \end{tabular}
IV. Find the probability of $k$ defective goods being contained within all shipped goods under the same conditions as in Question III.
V. Suppose that $P _ { i } = 0.001 i$ in five factories $i ( i = 1,2,3,4,5 )$ and the same number ($N _ { c}$) of goods are shipped from all these factories.
Find the maximum value of $N _ { c }$ for which the expected number of defective goods out of all shipped goods is equal to or less than 3.