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

2019 centrale-maths2__official

37 maths questions

Q1 Probability Generating Functions Characteristic function product or trigonometric identity View

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_X(t) = \mathbb{E}(\mathrm{e}^{\mathrm{i}tX})$.
Show $$\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right).$$

Q2 Probability Generating Functions Characteristic function product or trigonometric identity View

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$.
Deduce $$\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}.$$

Q3 Discrete Random Variables Convergence of Expectations or Moments View

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $$\operatorname{sinc}\, t = \begin{cases} \frac{\sin t}{t} & \text{if } t \neq 0 \\ 1 & \text{otherwise} \end{cases}$$
Determine the pointwise limit of the sequence of functions $(\Phi_{X_n})_{n \geqslant 1}$.

Q4 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$.
Study the continuity of $\lim_{n \rightarrow +\infty} \Phi_{X_n}$.

Q5 Discrete Random Variables Independence Proofs for Discrete Random Variables View

Let $n$ be a non-zero natural number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$.
Show that $X_n$ and $-X_n$ have the same distribution for all $n \in \mathbb{N}^{\star}$.

Q6 Moment generating functions Pointwise limit of MGFs or characteristic functions (convergence in distribution) View

Let $n$ be a non-zero natural number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$.
Deduce the pointwise limit of the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}(\cos(t X_n)) \end{aligned}$$

Q7 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $n$ be a non-zero natural number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}(\cos(t X_n)) \end{aligned}$$
Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ converge uniformly on $\mathbb{R}$?

Q8 Number Theory Combinatorial Number Theory and Counting View

Let $n$ be a non-zero natural number. We set $$\Phi_n : \left|\, \begin{aligned} \{0,1\}^n &\rightarrow \llbracket 0, 2^n - 1 \rrbracket \\ (x_j)_{j \in \llbracket 1,n \rrbracket} &\mapsto \sum_{j=1}^{n} x_j 2^{n-j} \end{aligned} \right.$$
Show that $\Phi_n$ is well-defined by verifying $\operatorname{Im} \Phi_n \subset \llbracket 0, 2^n - 1 \rrbracket$.

Q9 Proof Direct Proof of a Stated Identity or Equality View

Let $n$ be a non-zero natural number. We set $$\Phi_n : \left|\, \begin{aligned} \{0,1\}^n &\rightarrow \llbracket 0, 2^n - 1 \rrbracket \\ (x_j)_{j \in \llbracket 1,n \rrbracket} &\mapsto \sum_{j=1}^{n} x_j 2^{n-j} \end{aligned} \right.$$ and $A_n = \left\{ \sum_{j=1}^{n} x_j 2^{n-j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Specify $\operatorname{Im} \Phi_n$ as a function of $A_n$.

Q10 Proof Proof by Induction or Recursive Construction View

Let $n$ be a non-zero natural number. We set $$\Phi_n : \left|\, \begin{aligned} \{0,1\}^n &\rightarrow \llbracket 0, 2^n - 1 \rrbracket \\ (x_j)_{j \in \llbracket 1,n \rrbracket} &\mapsto \sum_{j=1}^{n} x_j 2^{n-j} \end{aligned} \right.$$
Show by induction $$\forall k \in \llbracket 0, 2^n - 1 \rrbracket, \quad k \in \operatorname{Im} \Phi_n.$$

Q11 Proof Deduction or Consequence from Prior Results View

Q12 Sequences and Series Evaluation of a Finite or Infinite Sum View

We denote $$D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\} \quad \text{and} \quad D = \bigcup_{n \in \mathbb{N}^{\star}} D_n.$$
Establish the monotonicity in the sense of inclusion of the sequence $(D_n)_{n \geqslant 1}$ then verify $D \subset [0,1[$.

Q13 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

For all $(x,n) \in \mathbb{R} \times \mathbb{N}$, we define $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$.
Establish $$\forall (x,n) \in \mathbb{R} \times \mathbb{N}, \quad \pi_n(x) \leqslant x < \pi_n(x) + \frac{1}{2^n}.$$

Q14 Sequences and Series Evaluation of a Finite or Infinite Sum View

For all $(x,n) \in \mathbb{R} \times \mathbb{N}$, we define $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ and $d_{n+1}(x) = 2^{n+1}(\pi_{n+1}(x) - \pi_n(x))$.
Justify $$\forall x \in [0,1[,\, \forall k \in \mathbb{N}, \quad \pi_k(x) = \sum_{j=1}^{k} \frac{d_j(x)}{2^j}.$$

Q15 Sequences and Series Recurrence Relations and Sequence Properties View

For all $(x,n) \in \mathbb{R} \times \mathbb{N}$, we define $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ and $d_{n+1}(x) = 2^{n+1}(\pi_{n+1}(x) - \pi_n(x))$.
Establish $$\forall (x,j) \in \mathbb{R} \times \mathbb{N}^{\star}, \quad d_j(x) \in \{0,1\}.$$

Q16 Sequences and Series Evaluation of a Finite or Infinite Sum View

We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$ and $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$.
Let $n \in \mathbb{N}^{\star}$. Justify $x \in D_n \Longleftrightarrow 2^n x \in \llbracket 0, 2^n - 1 \rrbracket$.

Q17 Proof Proof That a Map Has a Specific Property View

We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Let $n \in \mathbb{N}^{\star}$. Show that the application $$\Psi_n : \begin{gathered} \{0,1\}^n \rightarrow D_n \\ (x_j)_{j \in \llbracket 1,n \rrbracket} \mapsto \sum_{j=1}^{n} \frac{x_j}{2^j} \end{gathered}$$ is bijective.

Q18 Sequences and Series Evaluation of a Finite or Infinite Sum View

We denote $\pi_k(x) = \frac{\lfloor 2^k x \rfloor}{2^k}$.
Let $n \in \mathbb{N}^{\star}$ and $x = \sum_{j=1}^{n} \frac{x_j}{2^j}$ with $(x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n$. Show $$\forall k \in \mathbb{N}, \quad \pi_k(x) = \sum_{j=1}^{\min(n,k)} \frac{x_j}{2^j}.$$

Q19 Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables View

Q20 Cumulative distribution functions View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x).$$ We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Show $$\forall n \in \mathbb{N}^{\star},\, \forall x \in D_n, \quad F_n(x) = x + \frac{1}{2^n}.$$

Q21 Cumulative distribution functions View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad G_n(x) = \mathbb{P}(Y_n < x).$$ We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Show $$\forall n \in \mathbb{N}^{\star},\, \forall x \in D_n, \quad G_n(x) = x.$$

Q22 Uniform Distribution View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Establish, for every non-zero natural number $n$, that $Y_n$ follows a uniform distribution on $D_n$.

Q23 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View

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 $$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$

Q24 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$
Let $x$ be a real number. Establish the monotonicity of the sequences $(F_n(x))_{n \geqslant 1}$ and $(G_n(x))_{n \geqslant 1}$.

Q25 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$
Deduce the pointwise convergence of the sequences of functions $(F_n)_{n \geqslant 1}$ and $(G_n)_{n \geqslant 1}$.

Q26 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$ We denote $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ where $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Show $$\forall x \in D \cup \{1\}, \quad \lim_{n \rightarrow \infty} F_n(x) = x \quad \text{and} \quad \lim_{n \rightarrow \infty} G_n(x) = x.$$

Q27 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$ We denote $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ where $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Generalize the results obtained in the previous question for all $x \in [0,1]$.

Q28 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Q29 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View

Q30 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ For every continuous function $f$ from $[0,1]$ to $\mathbb{R}$, the sequence $(\mathbb{E}(f(Y_n)))_{n \geqslant 1}$ converges to $\int_0^1 f(t)\,\mathrm{d}t$.
Using the previous result, propose another proof of the result obtained in question 6.

Q31 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ For every continuous function $f$ from $[0,1]$ to $\mathbb{R}$, the sequence $(\mathbb{E}(f(Y_n)))_{n \geqslant 1}$ converges to $\int_0^1 f(t)\,\mathrm{d}t$.
An application. Justify the existence of $\int_0^1 \frac{t-1}{\ln t}\,\mathrm{d}t$ then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n})\,\mathrm{d}t$.

Q32 Number Theory Combinatorial Number Theory and Counting View

We denote $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ where $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Is the set $D$ countable?

Q33 Proof Existence Proof View

Suppose that there exists $f : \mathbb{N} \rightarrow \mathcal{P}(\mathbb{N})$ bijective. By considering $A = \{x \in \mathbb{N} \mid x \notin f(x)\}$, establish a contradiction.

Q34 Proof Proof That a Map Has a Specific Property View

Show that the application $\Phi : \begin{aligned} \mathcal{P}(\mathbb{N}) &\rightarrow \{0,1\}^{\mathbb{N}} \\ A &\mapsto \mathbb{1}_A \end{aligned}$ is bijective.

Q35 Proof Proof That a Map Has a Specific Property View

Show that the application $$\Psi : \begin{aligned} \{0,1\}^{\mathbb{N}} &\rightarrow [0,1] \\ (x_n) &\mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}} \end{aligned}$$ is well-defined and surjective. Is it injective?

Q36 Proof Proof That a Map Has a Specific Property View

We denote $D^{\star} = D \setminus \{0\}$ where $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ and $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$. We set for all $(x_n) \in \{0,1\}^{\mathbb{N}}$ $$\Lambda\left((x_n)\right) = \begin{cases} \Psi\left((x_n)\right) & \text{if } \Psi\left((x_n)\right) \in [0,1[ \setminus D^{\star} \\ \frac{\Psi\left((x_n)\right)}{2} & \text{if } \Psi\left((x_n)\right) \in D \cup \{1\} \text{ and } (x_n) \text{ is eventually constant at } 1 \\ \frac{1 + \Psi\left((x_n)\right)}{2} & \text{if } \Psi\left((x_n)\right) \in D^{\star} \text{ and } (x_n) \text{ is eventually constant at } 0 \end{cases}$$ where $\Psi : \{0,1\}^{\mathbb{N}} \rightarrow [0,1]$, $(x_n) \mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}}$.
Show that $\Lambda$ realizes a bijection from $\{0,1\}^{\mathbb{N}}$ to $[0,1[$.

Q37 Number Theory Combinatorial Number Theory and Counting View

Using the results of the previous questions (in particular that $\Lambda$ is a bijection from $\{0,1\}^{\mathbb{N}}$ to $[0,1[$, and that $\mathcal{P}(\mathbb{N})$ is not countable), conclude that $[0,1[$ is not countable.