Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Let $a , b \in \mathbf { R }$ such that $a \neq 0$ and $| b | \leq | a |$. Show that
$$| a + b | = | a | + \operatorname { sign } ( a ) b$$
where $\operatorname { sign } ( x ) = x / | x |$ for nonzero real $x$. Deduce that:
$$\forall n \in \mathbf { N } ^ { * } , \quad E \left( \left| S _ { 2 n } \right| \right) = E \left( \left| S _ { 2 n - 1 } \right| \right)$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$, and for all $x \in \mathbb{R}$, $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$.
Let $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$. Show that the random variable $\chi_B(x)$ has finite expectation, and that, denoting by $\chi_A'$ the derivative of the polynomial $\chi_A$, we have $$\mathbb{E}\left[\chi_B(x)\right] = \chi_A(x) - \frac{1}{n} \chi_A'(x)$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$, and for all $x \in \mathbb{R}$, $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$.
Show that for all $k \in \{1, 2, \ldots, n\}$, we have $$\mathbb{E}\left[\chi_B\left(\lambda_k\right)\right] = -\frac{1}{n} \chi_A'\left(\lambda_k\right)$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$, and for all $x \in \mathbb{R}$, $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$.
Prove that there exists $x \in \mathbb{R}$ such that $\mathbb{E}\left[\chi_B(x)\right] \neq 0$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. For all $x \in \mathbb{R}$, we denote $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$. Let $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$. Show that the random variable $\chi_B(x)$ has finite expectation, and that, denoting by $\chi_A'$ the derivative of the polynomial $\chi_A$, we have $$\mathbb{E}\left[\chi_B(x)\right] = \chi_A(x) - \frac{1}{n} \chi_A'(x).$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Show that for all $k \in \{1,2,\ldots,n\}$, we have $$\mathbb{E}\left[\chi_B\left(\lambda_k\right)\right] = -\frac{1}{n} \chi_A'\left(\lambda_k\right).$$