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 $P \left( \left| M _ { n } - 22 \right| \geqslant 3 \right) < 0.01$ ''.

From a population following a normal distribution $\mathrm { N } \left( 50,8 ^ { 2 } \right)$, a sample of size 16 is randomly extracted to obtain the sample mean $\bar { X }$. From a population following a normal distribution $\mathrm { N } \left( 75 , \sigma ^ { 2 } \right)$, a sample of size 25 is randomly extracted to obtain the sample mean $\bar { Y }$. When $\mathrm { P } ( \bar { X } \leq 53 ) + \mathrm { P } ( \bar { Y } \leq 69 ) = 1$, what is the value of $\mathrm { P } ( \bar { Y } \geq 71 )$ using the standard normal distribution table below?

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.2	0.3849
1.4	0.4192
1.6	0.4452

[4 points]
(1) 0.8413
(2) 0.8644
(3) 0.8849
(4) 0.9192
(5) 0.9452

Let $\bar { X }$ be the sample mean obtained by randomly sampling 9 items from a population following the normal distribution $\mathrm { N } \left( 0,4 ^ { 2 } \right)$, and let $\bar { Y }$ be the sample mean obtained by randomly sampling 16 items from a population following the normal distribution $\mathrm { N } \left( 3,2 ^ { 2 } \right)$. What is the value of the constant $a$ satisfying $\mathrm { P } ( \bar { X } \geq 1 ) = \mathrm { P } ( \bar { Y } \leq a )$? [3 points]
(1) $\frac { 19 } { 8 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 21 } { 8 }$
(4) $\frac { 11 } { 4 }$
(5) $\frac { 23 } { 8 }$

For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$.
Deduce that $$\lim_{C \rightarrow +\infty} \sigma_{ij}(C) = 1$$

For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$. If $\sigma_{ij}(C) \neq 0$, we set $$\widehat{X}_{ij}(C) = \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)\right).$$
Justify that, for $C$ large enough, the variables $\widehat{X}_{ij}(C)$ are well defined and that they are then bounded, centered, of variance 1 and that they are mutually independent for $1 \leqslant i \leqslant j$.

For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$. If $\sigma_{ij}(C) \neq 0$, we set $$\widehat{X}_{ij}(C) = \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)\right).$$
Show that $$X_{ij} - \widehat{X}_{ij}(C) = \left(1 - \frac{1}{\sigma_{ij}(C)}\right) X_{ij} + \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| > C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| > C}\right)\right).$$

For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$. If $\sigma_{ij}(C) \neq 0$, we set $$\widehat{X}_{ij}(C) = \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)\right).$$
Show that $$\lim_{C \rightarrow +\infty} \mathbb{E}\left(\left(X_{ij} - \widehat{X}_{ij}(C)\right)^{2}\right) = 0.$$

For every integer $n$ such that $n \geqslant 1$, we denote by $\widehat{M}_{n}(C) = \left(\widehat{X}_{ij}(C)\right)_{1 \leqslant i,j \leqslant n}$. For every $\omega \in \Omega$, we denote by $\widehat{\Lambda}_{1,n}(\omega) \geqslant \ldots \geqslant \widehat{\Lambda}_{n,n}(\omega)$ the eigenvalues of $\frac{1}{\sqrt{n}} \widehat{M}_{n}(C)(\omega)$ arranged in decreasing order. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $K$-Lipschitz function.
Show that $$\left|\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) - \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\widehat{\Lambda}_{i,n}\right)\right)\right| \leqslant \frac{K}{n} \mathbb{E}\left(\left\|M_{n} - \widehat{M}_{n}(C)\right\|_{F}\right).$$

For every integer $n$ such that $n \geqslant 1$, we denote by $\widehat{M}_{n}(C) = \left(\widehat{X}_{ij}(C)\right)_{1 \leqslant i,j \leqslant n}$. For every $\omega \in \Omega$, we denote by $\widehat{\Lambda}_{1,n}(\omega) \geqslant \ldots \geqslant \widehat{\Lambda}_{n,n}(\omega)$ the eigenvalues of $\frac{1}{\sqrt{n}} \widehat{M}_{n}(C)(\omega)$ arranged in decreasing order. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $K$-Lipschitz function.
Assume furthermore that $f$ is bounded. Show $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) \xrightarrow{n \rightarrow +\infty} \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x.$$

Show the semicircle law in the general case:
For every function $f : \mathbb{R} \rightarrow \mathbb{R}$, continuous and bounded, $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) \xrightarrow{n \rightarrow +\infty} \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $U = \left(\begin{array}{c} u_1 \\ \vdots \\ u_n \end{array}\right)$ in $\mathcal{M}_{n,1}(\mathbb{R})$. We define the discrete random variable $X = U^\top Y$.
Show that $X$ admits a variance and that $$\mathbb{V}(X) = U^\top \Sigma_Y U.$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. The objective is to show that $$\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1.$$ We denote by $r$ the rank of the covariance matrix of $Y$.
Handle the case where $r = n$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. The objective is to show that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$. We now assume $r < n$ where $r$ is the rank of $\Sigma_Y$.
Prove that the kernel and image of $\Sigma_Y$ are supplementary orthogonal subspaces in $\mathcal{M}_{n,1}(\mathbb{R})$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$.
Prove that $$\forall j \in \llbracket 1, d \rrbracket, \quad \mathbb{V}\left(V_j^\top(Y - \mathbb{E}(Y))\right) = 0.$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$.
Deduce that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$, and we have shown that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$ for all $j \in \llbracket 1, d \rrbracket$.
Conclude that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$.

We set $A_2 = \operatorname{diag}(9, 5, 4)$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. In this question only, we assume that $Y$ is a random variable with values in $\mathcal{M}_{3,1}(\mathbb{R})$ such that $\Sigma_Y = A_2$. Determine the maximum of $q_Y$ on $C$, where $q_Y(U) = \mathbb{V}(U^\top Y)$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$ and $q_Y(U) = \mathbb{V}(U^\top Y)$.
In the general case, prove that the function $q_Y$ admits a maximum on $C$. Specify the value of this maximum as well as a vector $U_0 \in C$ such that $$\max_{U \in C} \mathbb{V}\left(U^\top Y\right) = \mathbb{V}\left(U_0^\top Y\right).$$

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1.
Prove that $\gamma \leqslant 1$ and express $\Sigma_Y$ in terms of $J$.

We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1.
Determine the eigenvalues of $J$ and the dimension of each associated eigenspace. Also determine an eigenvector associated with its eigenvalue of maximal modulus.