The table shows the probability distribution of a random variable X, which can only take the values $1,2,3,4$ and 5.

k	1	2	3	4	5
$\mathrm { P } ( \mathrm { X } = \mathrm { k } )$	$\mathrm { p } _ { 1 }$	$\mathrm { p } _ { 2 }$	$\mathrm { p } _ { 3 }$	0,2	0,15

The probabilities $P ( X = 4 )$ and $P ( X = 5 )$ as well as the expected value and the variance of X are known. From this information, the following system of equations results, with which the missing probabilities $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ can be calculated.
I $p _ { 1 } + p _ { 2 } + p _ { 3 } = 0,65$ II $p _ { 1 } + 2 p _ { 2 } + 3 p _ { 3 } = 1,45$ III $\quad 4 p _ { 1 } + p _ { 2 } = 0,6$ Determine, without solving the system of equations, which values for the expected value and the variance of X were used when setting up the system of equations.

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 $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Give the distribution, expectation and variance of the random variables $\operatorname{tr}(M)$ and $\operatorname{rg}(M)$.

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 $S = X_1 + \ldots + X_n$, $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Express $M^k$ in terms of $S$ and $M$.
What is the probability that the sequence of matrices $(M^k)_{k \in \mathbb{N}}$ is convergent?
Show that, in this case, the limit is a projection matrix.

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$.
For an integer $k \geqslant 1$, give the value of $P(T_{i,j} \geqslant k)$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$.
Determine the expectation and variance of $S_{n}$.

Let $S$ and $T$ be two finite real random variables that are independent and defined on $(\Omega, \mathcal{A}, P)$. We assume that $T$ and $-T$ have the same distribution.
Show that $E(\cos(S + T)) = E(\cos(S)) E(\cos(T))$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$.
We consider the function $\varphi_{n}$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\varphi_{n}(t) = E\left(\cos\left(S_{n} t\right)\right)$ for all real $t$.
Show that $\varphi_{n}(t) = (\cos t)^{n}$ for all integers $n \in \mathbb{N}^{*}$ and all real $t$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, and $u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$.
Show, for all $n \in \mathbb{N}^{*}$, $$E\left(\left|S_{n}\right|\right) = \frac{2}{\pi} u_{n}$$ We will use the integral expression for the absolute value obtained in question I.A.5.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$ and $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$.
Show that $E\left(S_{n}^{4}\right) = 3n^{2} - 2n$ for all $n \in \mathbb{N}^{*}$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$ and $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$.
Show that, for all $n \in \mathbb{N}^{*}$, $$P\left(U_{n} \geqslant \frac{1}{\sqrt{n}}\right) \leqslant \frac{3}{n^{3/2}}$$

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$, and $$\mathcal{Z}_{n} = \left\{\omega \in \Omega, \exists k \geqslant n, U_{k}(\omega) \geqslant \frac{1}{\sqrt{k}}\right\}$$ Show that $\mathcal{Z}_{n} \in \mathcal{A}$ for all $n \in \mathbb{N}^{*}$ and that $\lim_{n \rightarrow \infty} P\left(\mathcal{Z}_{n}\right) = 0$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$, and $$\mathcal{Z}_{n} = \left\{\omega \in \Omega, \exists k \geqslant n, U_{k}(\omega) \geqslant \frac{1}{\sqrt{k}}\right\}$$ By considering $Z = \bigcap_{n \in \mathbb{N}^{*}} \mathcal{Z}_{n}$, show that $\left(\frac{S_{n}}{n}\right)$ converges almost surely to $0$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
Show that the sequence $\left(E\left(\left|T_{n}\right|\right)\right)_{n \in \mathbb{N}^{*}}$ is increasing.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
Show that if the series $\sum a_{n}^{2}$ is convergent, then the sequence $\left(E\left(\left|T_{n}\right|\right)\right)_{n \in \mathbb{N}^{*}}$ is convergent.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
We assume $a_{1} \geqslant a_{2} + \cdots + a_{n}$. Show $E\left(\left|T_{n}\right|\right) = E\left(\left|T_{1}\right|\right) = a_{1}$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $\left(J_{n}\right)_{n \in \mathbb{N}^{*}}$ is a well-defined sequence and that it is increasing and convergent.
We will set $a_{k} = \frac{1}{2k-1}$ and express the expectation of $\left|T_{n}\right|$ using the method of question II.A.4.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $J_{n} = \frac{\pi}{2}$ for $1 \leqslant n \leqslant 7$ and that $\left(J_{n}\right)_{n \geqslant 7}$ is strictly increasing.

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if $Y$ admits a finite expectation, that is, if the series $\sum n ^ { m } P ( Y = n )$ converges. We then call the moment of order $m$ of $Y$ the real number $$\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$$
Show that if $Y : \Omega \rightarrow \mathbb { N }$ is a random variable associated with a generating function $G _ { Y }$ of radius strictly greater than 1, then $Y$ admits a finite moment of all orders.

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We define $H_k(X) = X(X-1)\cdots(X-k+1)$ for $k \in \mathbb{N}^*$ and $H_0(X)=1$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if the series $\sum n ^ { m } P ( Y = n )$ converges, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
Let $Y : \Omega \rightarrow \mathbb { N }$ be a random variable admitting a finite moment of all orders.
IV.B.1) Show that the generating function $G _ { Y }$ is of class $C ^ { \infty }$ on $[ - 1,1 ]$.
IV.B.2) Express $G _ { Y } ^ { ( k ) } ( 1 )$ using the polynomials $H _ { k } ( X )$ and the random variable $Y$.
IV.B.3) Does the generating function $G _ { Y }$ necessarily have a radius of convergence strictly greater than 1? One may use the power series $\sum \mathrm { e } ^ { - \sqrt { n } } x ^ { n }$.

In this subsection, $n$ is a natural integer greater than or equal to 2 and $X$ is a random variable taking values in $\mathbb{N}$, defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and following the uniform distribution on $\llbracket 0, n-1 \rrbracket$: $$\mathbb{P}(X = k) = \frac{1}{n} \text{ if } k \in \llbracket 0, n-1 \rrbracket \text{ and } \mathbb{P}(X = k) = 0 \text{ otherwise}$$
We assume in this question that $n$ is not prime: there exist integers $a$ and $b$, greater than or equal to 2, such that $n = ab$.
a) Show that there exists a unique pair of integer-valued random variables $(Q, R)$ defined on $\Omega$ such that $$X = aQ + R \quad \text{and} \quad \forall \omega \in \Omega, R(\omega) \in \llbracket 0, a-1 \rrbracket$$ One may consider a Euclidean division.
b) Specify the distribution of $(Q, R)$, then the distributions of $Q$ and $R$.
c) Show that $X$ is decomposable. Deduce an expression of $G_{X}$ as a product of two non-constant polynomials that one will specify.

In this subsection, $n$ is a natural integer greater than or equal to 2 and $X$ is a random variable taking values in $\mathbb{N}$, defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and following the uniform distribution on $\llbracket 0, n-1 \rrbracket$: $$\mathbb{P}(X = k) = \frac{1}{n} \text{ if } k \in \llbracket 0, n-1 \rrbracket \text{ and } \mathbb{P}(X = k) = 0 \text{ otherwise}$$
We assume in this question that $n$ is a prime number and we establish that $X$ is not decomposable.
a) Show that it suffices to prove the following result: if $U$ and $V$ are monic polynomials in $\mathbb{R}[T]$ with coefficients in $\mathbb{R}_{+}$ such that $U(T)V(T) = 1 + T + \cdots + T^{n-1}$, then one of the two polynomials $U$ or $V$ is constant.
In what follows, we fix polynomials $U$ and $V$ in $\mathbb{R}[T]$ that are monic with coefficients in $\mathbb{R}_{+}$ such that $$U(T)V(T) = 1 + T + \cdots + T^{n-1}$$ We set $r = \deg U$ and $s = \deg V$ and we assume by contradiction that $r$ and $s$ are non-zero.
b) Show that $U(T) = T^{r} U\left(\frac{1}{T}\right)$ and $V(T) = T^{s} V\left(\frac{1}{T}\right)$.
We then denote $U(T) = 1 + u_{1}T + \cdots + u_{r-1}T^{r-1} + T^{r}$ and $V(T) = 1 + v_{1}T + \cdots + v_{s-1}T^{s-1} + T^{s}$ with $r \leqslant s$ (if necessary by exchanging the roles of $U$ and $V$).
c) Show that $\forall k \in \llbracket 1, r \rrbracket, u_{k}v_{k} = 0$.
d) Deduce that $\forall k \in \llbracket 1, r \rrbracket, u_{k} \in \{0,1\}$ and $v_{k} \in \{0,1\}$.
e) Conclude.
One may first show that all coefficients of $V$ take values in $\{0,1\}$.

Assume that $X$ is constant equal to $a \in \mathbb{R}$. Show that $X$ is infinitely divisible.

Let $X$ be a bounded infinitely divisible random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We denote $M = \sup_{\Omega} |X|$, so that $|X(\omega)| \leqslant M$ for all $\omega \in \Omega$.
Let $n \in \mathbb{N}^{*}$ and let $X_{1}, \ldots, X_{n}$ be independent random variables with the same distribution, such that $X_{1} + \cdots + X_{n}$ has the same distribution as $X$.
a) For all $i \in \llbracket 1, n \rrbracket$, show that $X_{i} \leqslant \frac{M}{n}$ almost surely, then $\left|X_{i}\right| \leqslant \frac{M}{n}$ almost surely.
b) Deduce that $\mathbb{V}(X) \leqslant \frac{M^{2}}{n}$, where $\mathbb{V}(X)$ denotes the variance of $X$.

Let $X$ be a bounded infinitely divisible random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We denote $M = \sup_{\Omega} |X|$, so that $|X(\omega)| \leqslant M$ for all $\omega \in \Omega$.
Conclude that $X$ is almost surely constant.

We introduce a uniformly distributed random variable $Z : \Omega \rightarrow \{-1,1\}^{n}$. For $\omega \in \Omega$, we denote by $Z_{i}(\omega)$ the coordinates of $Z(\omega)$. Show that for all $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$, we have $$\forall i \in \{1, \ldots, n\}, \quad \mathbb{E}\left[\left|\sum_{j=1}^{n} a_{i,j} Z_{j}\right|\right] = \frac{1}{2^{n}} \sum_{k=0}^{n} \binom{n}{k} |n - 2k|,$$ where $\binom{n}{k}$ denotes the binomial coefficient. Deduce $$\mathbb{E}\left[g_{A}(Z)\right] = \frac{n}{2^{n}} \sum_{k=0}^{n} \binom{n}{k} |n - 2k|.$$