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

2017 centrale-maths1__psi

17 maths questions

QI.A.1 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View

Let $U$ and $V$ be two random variables on $(\Omega, \mathcal{A}, P)$ possessing a second moment and such that $V$ is not almost surely zero. Show that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} \geqslant 0$ and that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} = 0$ if and only if there exists $\lambda \in \mathbb{R}$ such that $\lambda V + U$ is almost surely zero.

QI.A.2 Moment generating functions Existence and domain of the MGF View

a) Suppose that $X$ is bounded. Justify that $X$ satisfies $(C_{\tau})$ for all $\tau$ in $\mathbb{R}^{+*}$.
b) Suppose that $X$ follows the geometric distribution with parameter $p \in ]0,1[$ $$\forall k \in \mathbb{N}^{*}, \quad P(X = k) = p(1-p)^{k-1}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.
c) Suppose that $X$ follows the Poisson distribution with parameter $\lambda$: $$\forall k \in \mathbb{N}, \quad P(X = k) = \mathrm{e}^{-\lambda} \frac{\lambda^{k}}{k!} \quad \text{where } \lambda \in \mathbb{R}^{+*}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.

QI.A.3 Moment generating functions Existence and domain of the MGF View

Let $a$ and $b$ be two real numbers such that $a < b$. Suppose $E\left(\mathrm{e}^{aX}\right) < +\infty$ and $E\left(\mathrm{e}^{bX}\right) < +\infty$.
a) Show $\forall t \in [a,b]$, $\mathrm{e}^{tX} \leqslant \mathrm{e}^{aX} + \mathrm{e}^{bX}$. Deduce that $E\left(\mathrm{e}^{tX}\right) < +\infty$. What can we conclude about the set $\left\{t \in \mathbb{R} ; E\left(\mathrm{e}^{tX}\right) < +\infty\right\}$?
b) Let $k$ be in $\mathbb{N}$, $t$ in $]a,b[$. We denote by $\theta_{k,t,a,b}$ the function $y \in \mathbb{R} \mapsto \frac{y^{k} \mathrm{e}^{ty}}{\mathrm{e}^{ay} + \mathrm{e}^{by}}$. Determine the limits of $\theta_{k,t,a,b}$ at $+\infty$ and $-\infty$. Show that this function is bounded on $\mathbb{R}$.
c) Show that $E\left(|X|^{k} \mathrm{e}^{tX}\right) < +\infty$.
d) We return to the notations of question b). Let $k$ be in $\mathbb{N}$, $c$ and $d$ be two real numbers such that $a < c < d < b$. Show that there exists $M_{k,a,b,c,d} \in \mathbb{R}^{+}$ such that for all $t \in [c,d]$ and for all $y \in \mathbb{R}$: $\left|\theta_{k,t,a,b}(y)\right| \leqslant M_{k,a,b,c,d}$.

QI.A.4 Moment generating functions Existence and domain of the MGF View

In this question, $\tau$ is an element of $\mathbb{R}^{+*}$ and $X$ satisfies $(C_{\tau})$.
a) Show that the set of real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$ is an interval $I$ containing $[-\tau, \tau]$. For $t$ in $I$, we denote $\varphi_{X}(t) = E\left(\mathrm{e}^{tX}\right)$.
b) Show that if $X(\Omega)$ is finite, $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
c) Suppose now that $X(\Omega)$ is a countably infinite set. We denote $X(\Omega) = \left\{x_{n} ; n \in \mathbb{N}^{*}\right\}$ where $\left(x_{n}\right)_{n \in \mathbb{N}^{*}}$ is a sequence of pairwise distinct real numbers and we set for all $n \in \mathbb{N}^{*}$, $p_{n} = P\left(X = x_{n}\right)$. Using the results established in question I.A.3 and two theorems relating to series of functions which you will state completely, show that $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
d) Verify that for $t$ in the interior of $I$ and $k$ in $\mathbb{N}$, $\varphi_{X}^{(k)}(t) = E\left(X^{k} \mathrm{e}^{tX}\right)$.
e) Let $\psi_{X} = \frac{\varphi_{X}^{\prime}}{\varphi_{X}}$. Show that $\psi_{X}$ is increasing on $I$ and that, if $X$ is not almost surely equal to a constant, $\psi_{X}$ is strictly increasing on $I$.

QI.B.1 Central limit theorem View

Suppose that $X$ admits a second moment. Let $\delta$ be an element of $\mathbb{R}^{+*}$. Show that, for $n$ in $\mathbb{N}^{*}$, $$P\left(\left|S_{n} - nE(X)\right| \geqslant n\delta\right) \leqslant \frac{V(X)}{n\delta^{2}}$$

QI.B.2 Central limit theorem View

Suppose that $X$ admits a second moment. If $u$ and $v$ are two real numbers such that $u < E(X) < v$, determine the limit of the sequence $\left(\pi_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad \pi_{n} = P\left(nu \leqslant S_{n} \leqslant nv\right)$$

QI.C.1 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View

Let $\left(u_{n}\right)_{n \geqslant 0}$ be a real sequence such that: $\forall (m,n) \in \mathbb{N}^{2}, \quad u_{m+n} \geqslant u_{m} + u_{n}$. We suppose that the set $\left\{\frac{u_{n}}{n}, n \in \mathbb{N}^{*}\right\}$ is bounded above and we denote by $s$ its supremum.
Let $m$, $q$ and $r$ be elements of $\mathbb{N}$. We set $n = mq + r$. Compare the two real numbers $u_{n}$ and $qu_{m} + u_{r}$ and show that $u_{n} - ns \geqslant q\left(u_{m} - ms\right) + u_{r} - rs$.

QI.C.2 Sequences and series, recurrence and convergence Convergence proof and limit determination View

Let $\left(u_{n}\right)_{n \geqslant 0}$ be a real sequence such that: $\forall (m,n) \in \mathbb{N}^{2}, \quad u_{m+n} \geqslant u_{m} + u_{n}$. We suppose that the set $\left\{\frac{u_{n}}{n}, n \in \mathbb{N}^{*}\right\}$ is bounded above and we denote by $s$ its supremum.
We fix $m$ in $\mathbb{N}^{*}$ and $\varepsilon$ in $\mathbb{R}^{+*}$. Using the Euclidean division of $n$ by $m$, show that there exists an integer $N$ such that for all $n > N$, $$\frac{u_{n}}{n} \geqslant \frac{u_{m}}{m} - \varepsilon$$

QI.C.3 Sequences and series, recurrence and convergence Convergence proof and limit determination View

QII.A.1 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

Let $a$ be a real number. Show $P(X \geqslant a) = 0 \quad \Longleftrightarrow \quad \forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) = 0$.

QII.A.2 Discrete Random Variables Independence Proofs for Discrete Random Variables View

Let $a$ be a real number. Let $m$ and $n$ be in $\mathbb{N}$.
a) Show that $S_{m+n} - S_{m}$ and $S_{n}$ have the same distribution.
b) Let $b$ be a real number. Show $P\left(S_{m+n} \geqslant (n+m)b\right) \geqslant P\left(S_{n} \geqslant nb\right) P\left(S_{m} \geqslant mb\right)$.

QII.A.3 Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations View

Let $a$ be a real number. We suppose that $P(X \geqslant a) > 0$. Show that the sequence $\left(\frac{\ln\left(P\left(S_{n} \geqslant na\right)\right)}{n}\right)_{n \geqslant 1}$ is well-defined and admits a limit $\gamma_{a}$ that is negative or zero, satisfying $$\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\gamma_{a}}$$

QII.B.1 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

The interval $I$ and the function $\varphi_{X}$ are defined as in question I.A.4. We suppose that $X$ satisfies $(C_{\tau})$ for some $\tau > 0$, is not almost surely constant, and $a > E(X)$.
Show that, for $n$ in $\mathbb{N}^{*}$ and $t$ in $I \cap \mathbb{R}^{+}$ $$E\left(\mathrm{e}^{tS_{n}}\right) = \left(\varphi_{X}(t)\right)^{n}, \quad P\left(S_{n} \geqslant na\right) \leqslant \frac{\varphi_{X}(t)^{n}}{\mathrm{e}^{nta}}$$

QII.B.2 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

The interval $I$ and the function $\varphi_{X}$ are defined as in question I.A.4. We suppose that $X$ satisfies $(C_{\tau})$ for some $\tau > 0$, is not almost surely constant, and $a > E(X)$. We define the function $\chi : \begin{aligned} & I \rightarrow \mathbb{R} \\ & t \mapsto \ln\left(\varphi_{X}(t)\right) - ta \end{aligned}$
a) Show that the function $\chi$ is bounded below on $I \cap \mathbb{R}^{+}$. We denote by $\eta_{a}$ the infimum of $\chi$ on $I \cap \mathbb{R}^{+}$.
b) Give an equivalent of $\chi(t)$ as $t$ tends to 0. Deduce that $\eta_{a} < 0$.
c) Show $\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\eta_{a}}$. Deduce that $\gamma_{a} < 0$.
d) In each of the following two cases, determine the set of real numbers $a$ satisfying the conditions $P(X \geqslant a) > 0$ and $a > E(X)$; then, for $a$ satisfying these conditions, calculate $\eta_{a}$.
i. $X$ follows the Bernoulli distribution $\mathcal{B}(p)$ with $0 < p < 1$.
ii. $X$ follows the Poisson distribution $\mathcal{P}(\lambda)$ with $\lambda > 0$.

QII.C.1 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

We suppose here that the infimum $\eta_{a}$ of the function $\chi$ on $I \cap \mathbb{R}^{+}$ is attained at a point $\sigma$ interior to $I \cap \mathbb{R}^{+}$. Let $t$ be a real number interior to $I$ and such that $t > \sigma$, $b$ be a real number such that $b > \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}$.
a) Calculate $\sum_{x \in X(\Omega)} \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$.
We then admit (if necessary by modifying $(\Omega, \mathcal{A}, P)$) that there exists a random variable $X^{\prime}$ on $(\Omega, \mathcal{A})$ such that $X^{\prime}(\Omega) = X(\Omega)$ and whose probability distribution is given by $$\forall x \in X(\Omega), \quad P\left(X^{\prime} = x\right) = \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$$ and that there exists a sequence $\left(X_{n}^{\prime}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables defined on $(\Omega, \mathcal{A}, P)$ all following the same distribution as $X^{\prime}$.
b) Show $$E\left(X^{\prime}\right) = \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}, \quad E\left(X^{\prime}\right) > a$$

QII.C.2 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

We suppose here that the infimum $\eta_{a}$ of the function $\chi$ on $I \cap \mathbb{R}^{+}$ is attained at a point $\sigma$ interior to $I \cap \mathbb{R}^{+}$. Let $t$ be a real number interior to $I$ and such that $t > \sigma$, $b$ be a real number such that $b > \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}$. We admit that, if $n$ in $\mathbb{N}^{*}$ and if $f$ is a map from $X(\Omega)^{n}$ to $\mathbb{R}^{+}$, we have $$E\left(f\left(X_{1}^{\prime}, \ldots, X_{n}^{\prime}\right)\right) = \frac{E\left(f\left(X_{1}, \ldots, X_{n}\right) \mathrm{e}^{tS_{n}}\right)}{\varphi_{X}(t)^{n}}$$
a) For $n$ in $\mathbb{N}^{*}$, we set $S_{n}^{\prime} = \sum_{k=1}^{n} X_{k}^{\prime}$. Show $P\left(na \leqslant S_{n}^{\prime} \leqslant nb\right) \leqslant P\left(S_{n} \geqslant na\right) \frac{\mathrm{e}^{ntb}}{\varphi_{X}(t)^{n}}$.
$$\text{We may introduce the map } f : \left|\, \begin{array}{cl} X(\Omega)^{n} & \rightarrow \mathbb{R} \\ \left(x_{1}, \ldots, x_{n}\right) & \mapsto \begin{cases} 1 & \text{if } na \leqslant \sum_{i=1}^{n} x_{i} \leqslant nb \\ 0 & \text{otherwise} \end{cases} \end{array} \right.$$
b) Using questions I.B.2, II.B.2c and a) above, finally show that $\eta_{a} = \gamma_{a}$.

QII.C.3 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

In this question you may use the results from II.B.2d.
a) Let $\alpha$ be in $]0, 1/2[$. For $n$ in $\mathbb{N}^{*}$, we set $$A_{n} = \left\{k \in \{0, \ldots, n\}, \left|k - \frac{n}{2}\right| \geqslant \alpha n\right\}, \quad U_{n} = \sum_{k \in A_{n}} \binom{n}{k}$$ Determine the limit of the sequence $\left(U_{n}^{1/n}\right)_{n \geqslant 1}$.
b) Let $\lambda$ be in $\mathbb{R}^{+*}$, $\alpha$ be in $]\lambda, +\infty[$. For $n$ in $\mathbb{N}^{*}$, we set $$T_{n} = \sum_{\substack{k \in \mathbb{N} \\ k \geqslant \alpha n}} \frac{n^{k} \lambda^{k}}{k!}$$ Determine the limit of the sequence $\left(T_{n}^{1/n}\right)_{n \geqslant 1}$.