Let $X$ be the random variable giving the number of points scored by Victor during a match.
We admit that the expected value $E ( X ) = 22$ and the variance $V ( X ) = 65$. Victor plays $n$ matches, where $n$ is a strictly positive integer. Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ be the random variables giving the number of points scored during the $1 ^ { \text {st} } , 2 ^ { \mathrm { nd } } , \ldots , n$-th matches. We admit that the random variables $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are independent and follow the same distribution as $X$. We set $\quad M _ { n } = \frac { X _ { 1 } + X _ { 2 } + \ldots + X _ { n } } { n }$.

1. In this question, we take $n = 50$. a. What does the random variable $M _ { 50 }$ represent? b. Determine the expected value and variance of $M _ { 50 }$. c. Prove that $P \left( \left| M _ { 50 } - 22 \right| \geqslant 3 \right) \leqslant \frac { 13 } { 90 }$. d. Deduce that the probability of the event ``$19 < M _ { 50 } < 25$'' is strictly greater than 0.85.
2. Indicate, by justifying, whether the following statement is true or false: ``There is no natural number $n$ such that $P \left( \left| M _ { n } - 22 \right| \geqslant 3 \right) < 0.01$ ''.

The store has three identical automatic checkouts which, during a day, each triggered 20 checks. We denote $X_1, X_2$ and $X_3$ the random variables associating to each checkout the number of errors detected during this day. We admit that the random variables $X_1, X_2$ and $X_3$ are independent of each other and each follow a binomial distribution $\mathscr{B}(20; 0.165)$.

1. Determine the exact values of the expectation and variance of the random variable $X_1$.
2. We define the random variable $S$ by $S = X_1 + X_2 + X_3$.
   Justify that $E(S) = 9.9$ and that $V(S) = 8.2665$. For this question, we will use 10 as the value of $E(S)$. Using the Bienaymé-Chebyshev inequality, show that the probability that the total number of errors on the day is strictly between 6 and 14 is greater than 0.48.

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 $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$. Show that, for all $\varepsilon > 0$, there exists a real number $c(\varepsilon)$ such that, if $c \geqslant c(\varepsilon)$ and $n \in \mathbb{N}^{*}$, we have $\mathrm{P}\left(\left|T_{n}\right| \geqslant c\right) \leqslant \varepsilon$.

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 $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ For $k \in \mathbb{Z}$, we define $x_{k,n} = \frac{k - n\lambda}{\sqrt{n\lambda}}$. We consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$ for all $x \in \mathbb{R}$, which is $M$-Lipschitz for some $M > 0$.
a) Show that, if $x, h \in \mathbb{R}$ and $h > 0$, then $\left| hf(x) - \int_{x}^{x+h} f(t) \mathrm{d}t \right| \leqslant M \frac{h^{2}}{2}$.
b) Deduce from this, when $I_{n}$ is non-empty, an upper bound for $$\left| \frac{1}{\sqrt{n\lambda}} \sum_{k \in I_{n}} f\left(x_{k,n}\right) - \int_{x_{p,n}}^{x_{q+1,n}} f(t) \mathrm{d}t \right|$$ where $p$ is the smallest element of $I_{n}$ and $q$ is the largest.
c) Show that $$\lim_{n \rightarrow +\infty} \frac{1}{\sqrt{n\lambda}} \sum_{k \in I_{n}} f\left(x_{k,n}\right) = \int_{a}^{b} f(x) \mathrm{d}x$$

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 $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ For $k \in \mathbb{Z}$, we define $x_{k,n} = \frac{k - n\lambda}{\sqrt{n\lambda}}$. We consider the function $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$.
For all $k \in I_{n}$, we denote $y_{k,n} = \left(1 - \frac{x_{k,n}}{k}\sqrt{n\lambda}\right)^{k} \exp\left(x_{k,n}\sqrt{n\lambda}\right)$.
Let $\varepsilon > 0$. Prove the existence of an integer $N(\varepsilon)$ such that, for all $n \geqslant N(\varepsilon)$ and all $k \in I_{n}$, the following inequalities are satisfied:
a) $\frac{1-\varepsilon}{\sqrt{2\pi}} \frac{1}{\sqrt{n\lambda}} y_{k,n} \leqslant \mathrm{e}^{-n\lambda} \frac{(n\lambda)^{k}}{k!} \leqslant \frac{1+\varepsilon}{\sqrt{2\pi}} \frac{1}{\sqrt{n\lambda}} y_{k,n}$;
We will use Stirling's formula $n! \underset{n \rightarrow +\infty}{\sim} \sqrt{2\pi n} \left(\frac{n}{\mathrm{e}}\right)^{n}$.
b) $(1-\varepsilon) f\left(x_{k,n}\right) \leqslant y_{k,n} \leqslant (1+\varepsilon) f\left(x_{k,n}\right)$.

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 $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$
Express, in the form of an integral, $\lim_{n \rightarrow +\infty} \sum_{k \in I_{n}} \frac{(n\lambda)^{k}}{k!} \mathrm{e}^{-n\lambda}$.

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 $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.
Determine the limits, when $n \rightarrow +\infty$, of $$\mathrm{P}\left(T_{n} \geqslant a\right), \quad \mathrm{P}\left(T_{n} = a\right), \quad \mathrm{P}\left(T_{n} > a\right) \quad \text{and} \quad \mathrm{P}\left(T_{n} \leqslant b\right)$$

Let $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$. Using the results of question III.C.6), deduce the value of $\int_{-\infty}^{+\infty} f(x) \mathrm{d}x$.

Recall that a random variable $X$ follows the Poisson distribution $\mathcal{P}(\lambda)$ if $\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$ for all $n \in \mathbb{N}$. 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 $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.
Determine an equivalent, when $n \rightarrow +\infty$, of $$A_{n} = \sum_{k=0}^{\lfloor n\lambda \rfloor} \frac{(n\lambda)^{k}}{k!} \quad \text{and} \quad B_{n} = \sum_{k=\lfloor n\lambda \rfloor + 1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$$ where $\lfloor t \rfloor$ denotes the integer part of the real number $t$.
We will interpret $\mathrm{e}^{-n\lambda} A_{n}$ as the probability of an event related to $S_{n}$ and thus to $T_{n}$.

For $\lambda \neq 1$, we denote $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$ and $D_{n} = \sum_{k=n+1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$.
Determine $\lim_{n \rightarrow +\infty} \mathrm{e}^{-n\lambda} C_{n}$ if $\lambda < 1$ and $\lim_{n \rightarrow +\infty} \mathrm{e}^{-n\lambda} D_{n}$ if $\lambda > 1$.

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

Let $n$ be a strictly positive integer. We consider a family $(X_{k})_{1 \leqslant k \leqslant n}$ of $n$ random variables taking values in $\{1, \ldots, N\}$, pairwise independent and identically distributed, defined on a probability space $(\Omega, \mathscr{A}, \mathbf{P})$. We further assume that $\mathbf{P}(X_{1} = i) = p_{i}$ and that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. Show that for all $\epsilon > 0$, we have $\mathbf{P}\left(\left|\frac{1}{n} \ln\left(\prod_{k=1}^{n} p_{X_{k}}\right) + H_{N}(p)\right| \geqslant \epsilon\right)$ tends to 0 as $n$ tends to infinity.
Where $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$ and $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$

Suppose that $X$ admits a second moment. Let $\delta$ be an element of $\mathbb{R}^{+*}$. Show that, for $n$ in $\mathbb{N}^{*}$, $$P\left(\left|S_{n} - nE(X)\right| \geqslant n\delta\right) \leqslant \frac{V(X)}{n\delta^{2}}$$

Suppose that $X$ admits a second moment. If $u$ and $v$ are two real numbers such that $u < E(X) < v$, determine the limit of the sequence $\left(\pi_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad \pi_{n} = P\left(nu \leqslant S_{n} \leqslant nv\right)$$

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 variable $X$ has finite expectation. We will denote by $m$ the expectation of $X$.
b) Apply, with appropriate justifications, the weak law of large numbers to the sequence of random variables $\left(X_{k}\right)$.

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, and $m = \mathbb{E}(X)$.
Show that the sequence defined by: $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}-m\right| \geqslant \varepsilon\right)$ is bounded above by a sequence with limit zero and whose convergence rate is geometric. Compare this result to the upper bound obtained with the weak law of large numbers.

Deduce, using Stirling's formula, that there exists a real $\alpha \in ]0,1[$ such that $P\left(S_n > n\right) = O\left(\alpha^n\right)$.

Given a real $t > 0$, we set
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Show that $\sigma _ { t } \sim \frac { \pi } { \sqrt { 3 } t ^ { 3 / 2 } }$ as $t$ tends to $0 ^ { + }$. Deduce from this that, for all real $u$,
$$j ( t , u ) \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } e ^ { - u ^ { 2 } / 2 }$$

The function $B _ { n }$ is defined as in Q19, $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$, and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$.
Conclude that the sequence $\left( \Delta _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges to 0.

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. The functions $\varphi$ and $\Phi$ are defined by $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$ and $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$.
Deduce that we have $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \int _ { u } ^ { v } \varphi ( x ) \mathrm { d } x$$ then that $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \right\} \right) = 1 - \Phi ( u )$$

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
Show that there exists $x _ { 0 } \geqslant 1$ such that we have $$\forall x \geqslant x _ { 0 } , \quad \exists n _ { x } \in \mathbb { N } , \quad \forall n \geqslant n _ { x } , \quad x ^ { 2 } \mathbb { P } \left( \left\{ \left| S _ { n } \right| \geqslant x \sqrt { n } \right\} \right) \leqslant \varepsilon .$$

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
For $x _ { 0 }$ and $x$ as in the previous question, we fix $N \geqslant \frac { n _ { x } } { \varepsilon }$ and we choose $n \geqslant N$. Show that then $$x ^ { 2 } \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| S _ { p } \right| \geqslant 3 x \sqrt { n } \right\} \right) \leqslant 3 \varepsilon$$

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. We consider, on the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability, the random variable $X_{n}$ defined by $X_{n}(\sigma) = \omega(\sigma)$.
Justify that there exists a positive real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have $$\mathbb{P}\left(\left|X_{n} - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^{2} \ln(n)}$$

Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Justify that there exists a real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have $$\mathbb{P}\left(\left|X_n - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^2 \ln(n)}.$$