We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ defined on $(\Omega, \mathcal{B}, \mathbb{P})$ with real values and define the random vector $Y(\omega) = \left(\begin{array}{c} Y_1(\omega) \\ \vdots \\ Y_n(\omega) \end{array}\right)$. The covariance matrix $\Sigma_Y$ has general term $\sigma_{i,j} = \operatorname{cov}(Y_i, Y_j)$.
Verify that $\Sigma_Y$ is a symmetric matrix, that $$\Sigma_Y = \mathbb{E}\left((Y - \mathbb{E}(Y))(Y - \mathbb{E}(Y))^\top\right)$$ and that, if $U$ is a constant vector in $\mathcal{M}_{n,1}(\mathbb{R})$, then $$\Sigma_{Y+U} = \Sigma_Y.$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $p \in \mathbb{N}^*$ and $M \in \mathcal{M}_{p,n}(\mathbb{R})$. We define the discrete random variable $Z = MY$, with values in $\mathcal{M}_{p,1}(\mathbb{R})$. Justify that $Z$ admits an expectation and express $\mathbb{E}(Z)$ in terms of $\mathbb{E}(Y)$. Show that $Z$ admits a covariance matrix $\Sigma_Z$ and that $$\Sigma_Z = M \Sigma_Y M^\top.$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y$, and $\Sigma_X$ is a diagonal matrix.
Deduce that the eigenvalues of $\Sigma_Y$ are all positive.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y = \left(\begin{array}{c} X_1 \\ \vdots \\ X_n \end{array}\right)$.
Prove that the total variance of $X$ is equal to that of $Y$.

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix whose diagonal coefficients $\lambda_i$ are all positive. Prove the existence of a discrete random variable $Z$ with values in $\mathcal{M}_{n,1}(\mathbb{R})$ such that $\Sigma_Z = D$.

Let $A \in \mathcal{S}_n(\mathbb{R})$ be a symmetric matrix whose eigenvalues are positive. Prove the existence of a discrete random variable $Y$ with values in $\mathcal{M}_{n,1}(\mathbb{R})$ such that $\Sigma_Y = A$.

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 $\Sigma_Y$. 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 $A_2 = \operatorname{diag}(9, 5, 4)$. Justify the existence of a random vector whose covariance matrix is $A_2$.

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 $$\max_{U \in C} \mathbb{V}\left(U^\top Y\right) = \mathbb{V}\left(U_0^\top Y\right).$$