Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$.
Calculate the probability that $X_1, \ldots, X_n$ are all equal.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
What is the probability that $M$ has two distinct eigenvalues?

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. We consider a function $h$ uniformly continuous and bounded on $\mathbb{R}$ such that for all $x \in \mathbb{R}$, $h(x) \leqslant h(0)$ and $$h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$$ We recall that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Show that for all $n \in \mathbb{N}$ and $x \geqslant 0$ such that $\mathbb{P}\left(S_n = x\right) > 0$, we have $h(-x) = h(0)$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote, for every integer $n \geqslant 1$, $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ do not have the same parity, then $\mathbb{P}\left(S_{n} = k\right) = 0$.

Let $x \in \mathbb{R}$ such that $x > 1$. Show that we define the probability distribution of a random variable $X$ taking values in $\mathbb{N}^{*}$ by setting $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $(q_{1}, \ldots, q_{n}) \in \mathcal{P}^{n}$ be an $n$-tuple of distinct prime numbers. Show that the events $(X \in q_{1}\mathbb{N}^{*}), \ldots, (X \in q_{n}\mathbb{N}^{*})$ are mutually independent.

Let $(\ell_k)_{k \in \mathbb{N}^*}$ be the limits defined in Q33, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
Using the result of Q33, deduce that $(\ell_{k})_{k \in \mathbb{N}^{*}}$ defines a probability distribution on $\mathbb{N}^{*}$.

Let $n$ be a non-zero natural number and let $X_n$ be a random variable that follows a uniform distribution on $D_n$. Show that there exist random variables $V_1, \ldots, V_n$ mutually independent, each following a Bernoulli distribution with parameter $1/2$, and such that $$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$

We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Conversely, let $n$ be a non-zero natural number and let $X_n$ be a random variable that follows a uniform distribution on $D_n$. Show that there exist random variables $V_1, \ldots, V_n$ mutually independent, each following a Bernoulli distribution with parameter $1/2$, and such that $$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$

We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ For $1 \leqslant i < j \leqslant pA$, prove that the random variable $Y_i Y_j$ follows a Bernoulli distribution whose parameter we will specify.

We consider the graph $G$ represented in Figure 2. We denote $S _ { 1 } = \{ 1,3,6,8 \}$ and $S _ { 2 } = \{ 2,4,5,7 \}$.
Show that, if the point is on a vertex of part $S _ { 1 }$ at a given step, it will be on a vertex of part $S _ { 2 }$ at the next step and that, if it is on a vertex of $S _ { 2 }$ at a given step, it will be on a vertex of $S _ { 1 }$ at the next step.

If $X$ follows the distribution $\mathcal { R }$ (where $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$), specify the distribution of the random variable $\frac { 1 } { 2 } ( X + 1 )$.

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 $X Y$.

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 $\mathbb { P } \left( \left( C , C ^ { \prime } \right) \text { is linearly dependent} \right)$.

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 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 $\frac{1}{2}$.

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 $Y_n(\sigma) = \varepsilon(\sigma)$.
Specify the distribution of $Y_n$.