In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the results of questions 16 and 19, determine the expectation of $X_{n}$.

We assume that the random variables $X_{ij}$ are uniformly bounded: $\exists K \in \mathbb{R} \; \forall (i,j) \in (\mathbb{N}^{\star})^{2} \; |X_{ij}| \leqslant K$.
Let $k$ be a natural integer. Justify that the random variable $\sum_{i=1}^{n} \Lambda_{i,n}^{k}$ admits an expectation and that $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$

Let $A > 2$. Show that, for every $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$

Calculate the expectation and the variance of a variable following the distribution $\mathcal { R }$, where $\mathcal{R}$ is defined by $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$.

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 $\Sigma_Y = A$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Determine the average number of fixed points of a random permutation and its limit as $n$ tends to $+\infty$.

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. 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 $X_n(\sigma) = \omega(\sigma)$.
Prove that $\mathbb{E}\left[X_n\right] \underset{n \rightarrow +\infty}{=} \ln(n) + \gamma + O\left(\frac{1}{n}\right)$.

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Show that $$\frac{1}{n!} \sum_{k=1}^{n} k(k-1) s(n,k) = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^2}.$$

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. 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 $X_n(\sigma) = \omega(\sigma)$.
Deduce that $$\frac{1}{n!} \sum_{k=1}^{n} k^2 s(n,k) = \mathbb{E}\left[X_n\right] + \left(\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^2}\right).$$

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 $X_n(\sigma) = \omega(\sigma)$.
Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \omega(\sigma)^2 \underset{n \rightarrow +\infty}{=} (2\gamma+1)\ln(n) + c + \ln(n)^2 + O\left(\frac{\ln(n)}{n}\right)$$ for a real number $c$ to be determined.

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 $X_n(\sigma) = \omega(\sigma)$.
Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} (\omega(\sigma) - \ln(n))^2 \underset{n \rightarrow +\infty}{=} \ln(n) + c + O\left(\frac{\ln(n)}{n}\right).$$

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 $$\mathbb{P}\left(\left|X_n - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^2 \ln(n)}.$$

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 $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Determine, for all $n \in \mathbf { N } ^ { * }$, $E \left( S _ { n } \right)$ and $V \left( S _ { n } \right)$.

We assume that the random variables $X_1, \ldots, X_N$ are pairwise uncorrelated, that is: $$\forall 1 \leq m, n \leq N, \quad n \neq m \Rightarrow \mathbb{E}[X_n X_m] = 0.$$ Prove that $$\mathbb{E}\left[|S_N|^2\right] \leq N.$$ Deduce that, for all $t > 0$, $$\mathbb{P}\left(|S_N| > t\sqrt{N}\right) \leq \frac{1}{t^2}$$ where $S_N := X_1 + \cdots + X_N$.

Let $Z$ be a random variable taking values in $\{ 0 , \ldots , M \}$. Show that:
$$\mathbf { E } [ Z ] = \sum _ { ( s , i , r ) \in E } \left( \sum _ { k = 0 } ^ { M } k \mathbf { P } \left( Z = k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) \right) \mathbf { P } \left( \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right).$$

Justify that for all $n \geq 0$, the random variables $\tilde { S } _ { n } , \tilde { I } _ { n }$ and $\tilde { R } _ { n }$ as well as the random variables $\Delta \tilde { S } _ { n } , \Delta \tilde { I } _ { n }$ and $\Delta \tilde { R } _ { n }$, have finite expectation.
Here $\Delta U_n = U_{n+1} - U_n$ and the random variables take values in $\{0, \ldots, M\}$.

141- All the data in the stem-and-leaf plot below are multiplied by 3, then 40 units are subtracted from each of them. What is the new mean of the data?

Stem	\multicolumn{4}{c}{Leaf}
8	0	1	5
9	2	4	6	7
10	0	0	3	4 8

(1) 245 (2) 245 (3) 250 (4) 255
(1) $240$ (2) $245$ (3) $250$ (4) $255$

142- In 12 statistical data, the total sum of all data is 72 and the sum of their squares is 480. What is the coefficient of variation of this data?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{9}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{2}{5}$

141- Based on the following frequency distribution table, what is the value of the coefficient of variation of the data $x$?

$x - 44$	$-3$	$-1$	$1$	$3$	$5$
Frequency	$4$	$7$	$5$	$3$	$1$

(1) $0.05$ (2) $0.08$ (3) $0.1$ (4) $0.2$

142- The scores of a technical skills test for two workers $A$ and $B$ are as follows:
$A: 15, 14, 15, 16, 17, 19$
$B: 16, 14, 17, 14, 17, 18$
Which worker has greater precision?
(1) $A$ (2) $B$ (3) Equal (4) Unpredictable

141- To the statistical data shown in the frequency polygon, two data values $29$ and $32$ are added. What are the new median values?
[Figure: Frequency polygon with x-axis values 24, 27, 30, 33, 36 and y-axis (f) values approximately 8, 9, 15, 11, 12]
(1) $23$ (2) $24$
(3) $25$ (4) $26$
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142- If the mean of grouped data equals 16, to determine the fourth class frequency, which value is correct?

Class representative	12	14	16	18	20
Frequency	5	7	10	$a$	3

(1) $4/\Lambda\Delta$

(2) $4/9\Upsilon$

(3) $\Delta/\Delta\Delta$

(4) $\Delta/V4$