grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2022 x-ens-maths2__mp

40 maths questions

Q1a Sequences and Series Convergence/Divergence Determination of Numerical Series View
For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent, where $$g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right)$$
Q1b Proof Proof That a Map Has a Specific Property View
Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are odd.
Q1c Proof Proof That a Map Has a Specific Property View
Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are periodic with period 1.
Q1d Proof Proof That a Map Has a Specific Property View
Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.
Q2a Trig Proofs Trigonometric Identity Simplification View
Let $f(x) = \pi \operatorname{cotan}(\pi x)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x)$$
Q2b Sequences and Series Functional Equations and Identities via Series View
Let $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$g\left(\frac{x}{2}\right) + g\left(\frac{1+x}{2}\right) = 2g(x)$$
Q3a Proof Existence Proof View
Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.
Q3b Proof Existence Proof View
Let $D = f - g$ where $f(x) = \pi \operatorname{cotan}(\pi x)$ and $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and let $\widetilde{D}$ be its continuous extension to $\mathbb{R}$. Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that: $$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M$$
Q5a Taylor series Construct series for a composite or related function View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Using the identity $\pi x \operatorname{cotan}(\pi x) = 1 + 2\sum_{n=1}^{+\infty} \frac{x^2}{x^2 - n^2}$, show that: $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{x}{2}\operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}$$
Q5b Taylor series Construct series for a composite or related function View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Using the result $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{x}{2}\operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}$$ deduce: $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{ix}{e^{ix}-1} = 1 - \frac{ix}{2} - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} \cdot x^{2k}$$
Q6 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have $$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1}\zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right)$$
Q7a Taylor series Extract derivative values from a given series View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Show that the function $h$ is of class $C^\infty$ on $\mathbb{R}$ and that, for all $n \in \mathbb{N}^*$, we have $$h^{(2n)}(0) = \frac{(-1)^{n-1}(2n)!}{\pi^{2n} 2^{2n-1}} \zeta(2n)$$
Q7b Sequences and Series Functional Equations and Identities via Series View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Show that for all $n \in \mathbb{N}$: $$\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \geqslant 1 \end{cases}$$
Q7c Sequences and Series Evaluation of a Finite or Infinite Sum View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Using the relation $\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = 0$ for $n \geqslant 1$, calculate $b_2$, $b_4$ and $b_6$, then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.
Q8a Proof Proof of Set Membership, Containment, or Structural Property View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. We denote $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that $\mathscr{M}(E)$ is a subset of $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.
Q8b Proof Proof That a Map Has a Specific Property View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.
Q9 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and let $\mu$ be an element of $\mathscr{M}(E)$. Show that if the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in the normed vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$, then $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x)$$
Q10a Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x) \tag{1}$$ We also fix a real number $\varepsilon > 0$. Show that there exists a finite subset $F_\varepsilon$ of $E$ and an integer $N_\varepsilon \geqslant 0$ such that $\mu(F_\varepsilon) > 1 - \varepsilon$ and for all integer $n \geqslant N_\varepsilon$ $$\sum_{x \in F_\varepsilon} |\mu_n(x) - \mu(x)| < \varepsilon$$
Q10b Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying condition (1): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$. We fix $\varepsilon > 0$ and a finite subset $F_\varepsilon$ of $E$ and integer $N_\varepsilon$ as in 10a. Show that for every subset $A$ of $E$: $$\left|\mu_n(A) - \mu(A)\right| \leqslant \left|\mu_n(A \cap F_\varepsilon) - \mu(A \cap F_\varepsilon)\right| + \mu(E \backslash F_\varepsilon) + \mu_n(E \backslash F_\varepsilon)$$ and deduce that if $n \geqslant N_\varepsilon$, then $\left|\mu_n(A) - \mu(A)\right| < 4\varepsilon$.
Q10c Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Deduce that the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ if and only if it satisfies condition (1): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$.
Q11 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. For every integer $k \in \mathbb{N}^*$, we denote $\delta_k$ the probability measure on $E$ such that, for all $n \in \mathbb{N}^*$, $$\delta_k(\{x_n\}) = \begin{cases} 1 & \text{if } n = k \\ 0 & \text{otherwise} \end{cases}$$ Does the sequence $(\delta_k)_{k \in \mathbb{N}^*}$ converge in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$?
Q12a Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Show that there exists a sequence $(\varphi_k)_{k \in \mathbb{N}^*}$ of strictly increasing applications from $\mathbb{N}^*$ to $\mathbb{N}^*$ such that, for all $k \in \mathbb{N}^*$ and for all integer $1 \leqslant i \leqslant k$, the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ converges.
Q12b Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a. Show that for all $i \in \mathbb{N}^*$ and all integer $k \geqslant i$, the limit of the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ depends only on $i$ and not on $k$. We denote this limit $\mu_\infty(x_i)$.
Q12c Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a, with limits $\mu_\infty(x_i)$ as defined in 12b. Show that the application $$\begin{aligned} \psi : \mathbb{N}^* &\longrightarrow \mathbb{N}^* \\ k &\longmapsto \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{aligned}$$ is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.
Q12d Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $\mu_\infty$ as defined in 12b. Show that $\mu_\infty(x_i) \geqslant 0$ for all $i$ in $\mathbb{N}^*$, and that $\sum_{i=1}^{\infty} \mu_\infty(x_i) \leqslant 1$.
Q12e Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $\mu_\infty$ as defined in 12b. We say that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight if for all real $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of $E$ such that $\mu_n(F_\varepsilon) \geqslant 1 - \varepsilon$ for all natural integer $n$. Suppose further that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight. Show then that $\mu_\infty$ defines an element of $\mathscr{M}(E)$ which is a cluster point of the sequence $(\mu_n)_{n \in \mathbb{N}}$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.
Q13 Probability Definitions Proof of a Probability Identity or Inequality View
Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $E$. We call the law of the variable $X$ and we denote $\mu_X$ the application where $\{X \in A\} = \{\omega \in \Omega \text{ such that } X(\omega) \in A\}$. Verify that $\mu_X$ is a probability on $E$.
Q14 Proof Deduction or Consequence from Prior Results View
Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, \mathbf{P})$ and for all subset $A$ of $E$: $$\left|\mu_X(A) - \mu_Y(A)\right| \leqslant \mathbf{E}\left(\left|\mathbf{1}_{\{X \in A\}} - \mathbf{1}_{\{Y \in A\}}\right|\right)$$ and deduce that $\left\|\mu_X - \mu_Y\right\| \leqslant \mathbf{P}(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.
Q15a Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. We define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \longmapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$ Justify that the application $L$ is well defined.
Q15b Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined in 15a. Show that $\mathbf{P}(X_n \neq X) \leqslant \mathbf{P}(L \geqslant n)$ for all integer $n$ in $\mathbb{N}$.
Q15c Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View
Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Using the results of 14 and 15b, deduce that $\lim_{n \rightarrow +\infty} \left\|\mu_{X_n} - \mu_X\right\| = 0$.
Q16 Number Theory Divisibility and Divisor Analysis View
If $N \in \mathbb{N}^*$ and $p$ is a prime number, we denote $\nu_p(N)$ the $p$-adic valuation of $N$. For $n \in \mathbb{N}^*$, we define the application $$\psi_n : \mathbb{N}^* \longrightarrow \mathbb{N}^*, \quad x \longmapsto \prod_{i=1}^{n} p_i^{\nu_{p_i}(x)}$$ where $(p_i)_{i \in \mathbb{N}^*}$ is the sequence of prime numbers, ordered in increasing order. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Show that $$\forall x \in \mathbb{N}^*, \quad \mathbf{P}(X = x) = \lim_{n \rightarrow +\infty} \mathbf{P}(\psi_n(X) = x)$$
Q17a Proof Direct Proof of a Stated Identity or Equality View
Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. We recall that $(p_i)_{i \in \mathbb{N}^*}$ denotes the sequence of prime numbers, ordered in increasing order. Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\bigcup_{i=1}^{n+1} \mathbb{N}^* r p_i = \left(\bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) \cup \left(\mathbb{N}^* r p_{n+1} \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_{n+1} p_i\right)$$
Q17b Proof Proof by Induction or Recursive Construction View
Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the result of 17a, show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right)$$
Q17c Proof Deduction or Consequence from Prior Results View
Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.
Q18 Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:
  1. [i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
  2. [ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} \mathbf{P}(r \mid X_n) = \mathbf{P}(r \mid X)$.
Show that the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$.
Q19 Discrete Random Variables Properties of Probability Measures and Convergence of Measures View
Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$. For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $\mathbf{P}(r \mid X_n^{(i)})$ and show that $\mathbf{P}(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} \mathbf{P}\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}$$
Q20a Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
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 $k \in \mathbb{N}^*$.
Q20b Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
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 $\mu_s$.
Q21 Number Theory GCD, LCM, and Coprimality View
For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $s, n \in \mathbb{N}^*$ with $2 \leqslant s \leqslant n$. We randomly draw $s$ numbers from $\{1, 2, \ldots, n\}$ and we denote $P_n(s)$ the probability that these numbers are coprime. Show that $$\lim_{n \rightarrow +\infty} P_n(s) = \frac{1}{\zeta(s)}$$ and give the value of $\lim_{n \rightarrow +\infty} P_n(s)$ in the case where $s = 2$, then $s = 4$, and finally $s = 6$.