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

Let $X$ and $Y$ be two independent real random variables, each following the distribution $\mathcal { R }$ (where $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$). Determine the distribution of their product $X Y$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
For $s > 1$ fixed, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}.$$
Let $Z$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ that follows the distribution $\mu_s$. Calculate $P(k \mid Z)$ for $k \in \mathbb{N}^*$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
For $s > 1$ fixed, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}.$$
Let $s \geqslant 2$ be an integer. Let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$, and let $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ be their gcd.
Deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $Z$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ that follows the distribution $\mu_s$. Calculate $\mathbf{P}(k \mid Z)$ for $k \in \mathbb{N}^*$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $s \geqslant 2$ be an integer. Let $Z_n^{(s)}$ be the gcd of $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. Using the results of questions 18, 19, and 20a, deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.

In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$.
Calculate the probability of the event $M _ { 2 } \in \mathcal { N } _ { 2 }$.

In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$.
Calculate the probability of the event $M _ { 2 } \in \mathcal { G } \ell _ { 2 } ( \mathbb { R } )$.

We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$.
Let $\left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right) \in \{ - 1,1 \} ^ { n }$. Calculate $\mathbb { P } \left( \left( c _ { 1 } = \varepsilon _ { 1 } \right) \cap \cdots \cap \left( c _ { n } = \varepsilon _ { n } \right) \right)$.

We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$. We consider the random column matrices $C = \left( \begin{array} { c } c _ { 1 } \\ \vdots \\ c _ { n } \end{array} \right)$ and $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$.
Prove that, for all $\omega \in \Omega$, the family $\left( C ( \omega ) , C ^ { \prime } ( \omega ) \right)$ is linearly dependent if and only if there exists $\varepsilon \in \{ - 1,1 \}$ such that $C ^ { \prime } ( \omega ) = \varepsilon C ( \omega )$.

We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$. We consider the random column matrices $C = \left( \begin{array} { c } c _ { 1 } \\ \vdots \\ c _ { n } \end{array} \right)$ and $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$.
Deduce $\mathbb { P } \left( \left( C , C ^ { \prime } \right) \text { is linearly dependent} \right)$.

Let $n$ be a non-zero natural number.

1. Let $p$ be a vector projection of rank $r \in \mathbb { N }$.
   1. [1.1.] Give, as a function of $r$, a matrix $W$ of $p$ in an adapted basis.
   2. [1.2.] Give the possible spectra of $W$.
   3. [1.3.] Compare $\boldsymbol { \operatorname { rg } } ( W )$ and $\boldsymbol { \operatorname { tr } } ( W )$.
   4. [1.4.] Calculate $\boldsymbol { \operatorname { det } } ( W )$.

We consider the family $X _ { 1 } , \ldots , X _ { n }$ of independent random variables defined on the same probability space $( \Omega , \mathscr { A } , \mathbb { P } )$ all following the Bernoulli distribution with parameter $p \in ]0,1[$.
Let $M$ be a discrete random variable from $\Omega$ to $\mathscr { M } _ { n } ( \mathbb { R } )$ such that for all $\omega$ in $\Omega , M ( \omega )$ is diagonalisable and similar to $\Delta ( \omega ) = \operatorname { diag } \left( X _ { 1 } ( \omega ) , \ldots , X _ { n } ( \omega ) \right)$.

1. We denote by $T$ the random variable $\mathbf { tr } ( M )$.
   1. [2.1.] Determine $T ( \Omega )$, that is the set of values taken by the random variable $T$.
   2. [2.2.] Give the probability distribution of $T$ and the expectation of the random variable $T$.
2. Deduce the probability distribution of the random variable $R = \mathbf { rg } ( M )$.
3. We denote by $D$ the random variable $\boldsymbol { \operatorname { det } } ( M )$.
   1. [4.1.] Determine $D ( \Omega )$.
   2. [4.2.] Give the probability distribution of $D$ and calculate the expectation of the random variable $D$.
4. We propose to determine the probability of the event $Z$: ``the eigenspaces of the matrix $M$ all have the same dimension''.
   1. [5.1.] We denote by $V$ the event: ``$M$ has only one eigenvalue''. Calculate $\mathbb { P } ( V )$.
   2. [5.2.] Suppose $n$ is odd. Determine $\mathbb { P } ( Z )$.
   3. [5.3.] Suppose $n$ is even and set $n = 2 r$. Calculate $\mathbb { P } ( T = r )$. Deduce $\mathbb { P } ( Z )$.
5. For all $\omega \in \Omega$, we denote $U ( \omega ) = \left( \begin{array} { c } X _ { 1 } ( \omega ) \\ \vdots \\ X _ { n } ( \omega ) \end{array} \right) \in \mathscr { M } _ { n , 1 } ( \mathbb { R } )$ and $A ( \omega ) = U ( \omega ) \times ( U ( \omega ) ) ^ { \top } = \left( a _ { i j } ( \omega ) \right) _ { ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } }$.
   1. [6.1.] Let $\omega \in \Omega$. Determine, for all pairs $( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , a _ { i j } ( \omega )$.
   2. [6.2.] Give the probability distribution of each random variable $a _ { i j }$.
   3. [6.3.] Show that $\operatorname { tr } ( A ) = \sum _ { i = 1 } ^ { n } X _ { i }$.
   4. [6.4.] Determine the values taken by the random variable $\boldsymbol { \operatorname { rg } } ( A )$.
   5. [6.5.] For all $\omega$ in $\Omega$, give the eigenvalues of the matrix $A ( \omega )$.
   6. [6.6.] Determine the probability distribution of the random variable $\mathbf { rg } ( A )$.

Show that the distribution of the random variable $X_n$ is given by $$\forall k \in \llbracket 0, n \rrbracket \quad P_n\left(X_n = k\right) = \frac{1}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$

On the finite probability space $\left(\mathcal{S}_n, P_n\right)$, we define, for all $i \in \llbracket 1, n \rrbracket$, the random variable $U_i$ such that, for all $\sigma \in \mathcal{S}_n$, we have $U_i(\sigma) = 1$ if $\sigma(i) = i$, and $U_i(\sigma) = 0$ otherwise.
Show that $U_i$ follows a Bernoulli distribution with parameter $\frac{1}{n}$.
Show that, if $i \neq j$, the variable $U_i U_j$ follows a Bernoulli distribution whose parameter you will specify.

We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the random variable $Z_k$ representing the state of the system after $k$ impulses, with $Z_0$ being the certain variable with value 1. Let $n \in \mathbf{N}$. Let $j \in \llbracket 1;N \rrbracket$, show that $P(Z_n = j) = K^n[1,j]$. One may proceed by induction.

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 $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B_n$ is defined as in Q19.
Show that, for all $j \in \llbracket 0 , n \rrbracket$, $$\mathbb { P } \left( \left\{ T _ { n } = j \right\} \right) = \int _ { x _ { n , j } - 1 / \sqrt { n } } ^ { x _ { n , j } + 1 / \sqrt { n } } B _ { n } ( x ) \mathrm { d } x$$

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 $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Consider a pair $( u , v )$ of real numbers such that $u < v$, and denote $$J _ { n } = \left\{ j \in \llbracket 0 , n \rrbracket \left\lvert \, \frac { n + u \sqrt { n } } { 2 } \leqslant j \leqslant \frac { n + v \sqrt { n } } { 2 } \right. \right\}$$
Justify that $$\mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \sum _ { j \in J _ { n } } \mathbb { P } \left( \left\{ T _ { n } = j \right\} \right)$$

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_{n}, \mathscr{P}(\mathfrak{D}_{n}))$ equipped with the uniform probability. We define a random variable $Y_{n}$ by $Y_{n}(\sigma) = \varepsilon(\sigma)$.
Explicitly state the distribution of $Y_{n}$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_{n}, \mathscr{P}(\mathfrak{D}_{n}))$ equipped with the uniform probability. We define a random variable $Y_{n}$ by $Y_{n}(\sigma) = \varepsilon(\sigma)$.
Calculate, for all $\varepsilon \in \{-1, 1\}$, $\lim_{n \rightarrow +\infty} \mathbb{P}\left(Y_{n} = \varepsilon\right)$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_n, \mathscr{P}(\mathfrak{D}_n))$ equipped with the uniform probability. We define a random variable $Y_n$ by $Y_n(\sigma) = \varepsilon(\sigma)$.
Specify the distribution of $Y_n$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_n, \mathscr{P}(\mathfrak{D}_n))$ equipped with the uniform probability. We define a random variable $Y_n$ by $Y_n(\sigma) = \varepsilon(\sigma)$.
Calculate, for all $\varepsilon \in \{-1, 1\}$, $\lim_{n \rightarrow +\infty} \mathbb{P}(Y_n = \varepsilon)$.

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 $Z_{n}(\sigma) = \nu(\sigma)$.
Explicitly state the distribution of $Z_{n}$.

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 $Z_n(\sigma) = \nu(\sigma)$.
Specify the distribution of $Z_n$.

Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that $\mathbf{P}(X > 0) \leq \mathbf{E}(X)$.

We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. Show that for every $n \in \mathbf{N}^*$, the random variable $\frac{1+X_n}{2}$ follows a Bernoulli distribution with parameter $\frac{1}{2}$.