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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
2023 centrale-maths2__official

33 maths questions

Q1 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Show that $$I _ { n } \geqslant \frac { 1 } { 2 ^ { n } }.$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.
Q2 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Justify the existence of $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$ and give the exact value of $K _ { 1 }$.
Q3 Sequences and Series Estimation or Bounding of a Sum View
Show that $$\int _ { 1 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t = O \left( \frac { 1 } { n 2 ^ { n } } \right) .$$ One may lower bound $1 + t ^ { 2 }$ by a polynomial of degree 1.
Q4 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View
Deduce that, as $n$ tends to $+ \infty$, $$I _ { n } \sim K _ { n } .$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$ and $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.
Q5 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Establish the recurrence relation $K _ { n } = K _ { n + 1 } + \frac { 1 } { 2 n } K _ { n }$, where $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.
Q6 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
Using the recurrence relation $K _ { n } = K _ { n + 1 } + \frac { 1 } { 2 n } K _ { n }$ and the fact that $I_n \sim K_n$, deduce a simple equivalent of $I _ { n }$ as $n$ tends to $+ \infty$, where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.
Q7 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
Justify that $$\sqrt { n } I _ { n } = \int _ { 0 } ^ { \sqrt { n } } \frac { 1 } { \left( 1 + u ^ { 2 } / n \right) ^ { n } } \mathrm {~d} u$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.
Q8 Integration by Substitution Substitution to Evaluate Limit of an Integral Expression View
Show that $$\lim _ { n \rightarrow \infty } \sqrt { n } I _ { n } = \int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } } \mathrm {~d} u$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.
Q9 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
Deduce the values of $$\int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } } \mathrm {~d} u \quad \text { then of } \quad \int _ { - \infty } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } / 2 } \mathrm {~d} u .$$
Q10 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Let $x > 0$. By writing that $\varphi ( t ) \leqslant \frac { t } { x } \varphi ( t )$ for all $t \geqslant x$, show that $$\int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t \leqslant \frac { \varphi ( x ) } { x }$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Q11 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Let $x > 0$. Using the study of a well-chosen function, show that $$\frac { x } { x ^ { 2 } + 1 } \varphi ( x ) \leqslant \int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Q12 Normal Distribution Tail Bound or Concentration Inequality Proof View
Deduce a simple equivalent of $1 - \Phi ( x )$ as $x$ tends to $+ \infty$, where $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$ and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Q13 Probability Definitions Event Expression and Partition View
In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$. In the case where $n \geqslant 2$, define the events $$A _ { 1 } = \left\{ \left| R _ { 1 } \right| \geqslant 3 x \right\} \quad \text { and } \quad A _ { p } = \left\{ \max _ { 1 \leqslant i \leqslant p - 1 } \left| R _ { i } \right| < 3 x \right\} \cap \left\{ \left| R _ { p } \right| \geqslant 3 x \right\}$$ for $p \in \llbracket 2 , n \rrbracket$.
Express the event $A$ using the events $A _ { 1 } , A _ { 2 } , \ldots , A _ { n }$.
Q14 Probability Definitions Proof of a Probability Identity or Inequality View
In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$, and $A_1, \ldots, A_n$ as defined in Q13.
Show that we have $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \sum _ { p = 1 } ^ { n } \mathbb { P } \left( A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \right) .$$
Q15 Proof Proof of Set Membership, Containment, or Structural Property View
In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A_1, \ldots, A_n$ be as defined in Q13.
Justify that for all $p \in \llbracket 1 , n \rrbracket$, we have the inclusion $$A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \subset A _ { p } \cap \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} .$$
Q16 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$.
Deduce that $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} \right) .$$
Q17 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$.
Conclude that $$\forall x > 0 , \quad \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\} \right) \leqslant 3 \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { p } \right| \geqslant x \right\} \right) .$$
Q18 Proof Direct Proof of a Stated Identity or Equality View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Compare the real numbers $- x _ { n , k }$ and $x _ { n , n - k }$.
Q19 Curve Sketching Range and Image Set Determination View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Justify the existence of the real number $\Delta _ { n }$ for all $n \in \mathbb { N } ^ { * }$.
Q20 Normal Distribution Convergence in Distribution / Central Limit Theorem Application View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Show that, for all $n \in \mathbb { N } ^ { * }$, we have the equality $$\Delta _ { n } = \sup _ { x \geqslant 0 } \left| B _ { n } ( x ) - \varphi ( x ) \right| .$$
Q21 Curve Sketching Variation Table and Monotonicity from Sign of Derivative View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0.$$
For all $n \in \mathbb { N } ^ { * }$, show that $B _ { n }$ is a decreasing function on $\mathbb { R } ^ { + }$. One may distinguish according to whether $n$ is even or odd.
Q22 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. We introduce the set $I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$ and assume that $n$ and $k$ vary such that $k \in I _ { n }$.
Show that we have $$k ! ( n - k ) ! = 2 \pi \mathrm { e } ^ { - n } k ^ { k + 1 / 2 } ( n - k ) ^ { n - k + 1 / 2 } \left( 1 + O \left( \frac { 1 } { n } \right) \right)$$ as $n$ tends to infinity. One may use Stirling's formula: $n ! = \left( \frac { n } { \mathrm { e } } \right) ^ { n } \sqrt { 2 \pi n } \left( 1 + O \left( \frac { 1 } { n } \right) \right)$.
Q23 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19. We assume $k \in I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$.
Deduce that, as $n$ tends to $+ \infty$, we have $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \frac { 1 + O \left( \frac { 1 } { n } \right) } { \left( \frac { 2 k } { n } \right) ^ { k + 1 / 2 } \left( 2 - \frac { 2 k } { n } \right) ^ { n - k + 1 / 2 } }$$
Q24 Taylor series Limit evaluation using series expansion or exponential asymptotics View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19. We assume $k \in I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$.
Deduce that $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \frac { 1 + O \left( \frac { 1 } { n } \right) } { \left( 1 - \frac { x _ { n , k } ^ { 2 } } { n } \right) ^ { \frac { n + 1 } { 2 } } \left( 1 + \frac { x _ { n , k } } { \sqrt { n } } \right) ^ { \frac { x _ { n , k } } { 2 } \sqrt { n } } \left( 1 - \frac { x _ { n , k } } { \sqrt { n } } \right) ^ { - \frac { x _ { n , k } } { 2 } \sqrt { n } } }$$ then that $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \exp \left( - \frac { x _ { n , k } ^ { 2 } } { 2 } \right) \left( 1 + O \left( \frac { 1 } { \sqrt { n } } \right) \right)$$
Q25 Normal Distribution Convergence in Distribution / Central Limit Theorem Application View
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$. We fix $\varepsilon > 0$ and $\ell \in \mathbb{R}^+$ such that $\varphi(\ell) \leqslant \frac{\varepsilon}{2}$.
Show that there exists a natural number $n _ { 1 }$ such that, for all integers $n \geqslant n _ { 1 }$, $$\sup _ { x \in [ 0 , \ell ] } \left| B _ { n } ( x ) - \varphi ( x ) \right| \leqslant \frac { \varepsilon } { 2 }$$
Q26 Normal Distribution Tail Bound or Concentration Inequality Proof View
The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
For all $\ell > 0$, show that there exists a natural number $n _ { 2 }$, such that, for all $n \geqslant n _ { 2 }$, $$B _ { n } ( \ell ) \leqslant 2 \varphi ( \ell )$$
Q27 Central limit theorem View
The function $B _ { n }$ is defined as in Q19, $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$, and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$.
Conclude that the sequence $\left( \Delta _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges to 0.
Q28 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $I$ be an interval of $\mathbb { R }$ and $\left( f _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ a sequence of functions piecewise continuous on $I$ that converges uniformly on $I$ to a function $f$ also piecewise continuous on $I$.
If $\left( u _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ (respectively $\left( v _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$) is a sequence of real numbers belonging to $I$ that converges to $u \in I$ (respectively $v \in I$), show that $$\lim _ { n \rightarrow + \infty } \left( \int _ { u _ { n } } ^ { v _ { n } } f _ { n } ( x ) \mathrm { d } x \right) = \int _ { u } ^ { v } f ( x ) \mathrm { d } x$$
Q29 Discrete Probability Distributions Binomial Distribution Identification and Application View
Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B_n$ is defined as in Q19.
Show that, for all $j \in \llbracket 0 , n \rrbracket$, $$\mathbb { P } \left( \left\{ T _ { n } = j \right\} \right) = \int _ { x _ { n , j } - 1 / \sqrt { n } } ^ { x _ { n , j } + 1 / \sqrt { n } } B _ { n } ( x ) \mathrm { d } x$$
Q30 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Consider a pair $( u , v )$ of real numbers such that $u < v$, and denote $$J _ { n } = \left\{ j \in \llbracket 0 , n \rrbracket \left\lvert \, \frac { n + u \sqrt { n } } { 2 } \leqslant j \leqslant \frac { n + v \sqrt { n } } { 2 } \right. \right\}$$
Justify that $$\mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \sum _ { j \in J _ { n } } \mathbb { P } \left( \left\{ T _ { n } = j \right\} \right)$$
Q31 Central limit theorem View
Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. The functions $\varphi$ and $\Phi$ are defined by $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$ and $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$.
Deduce that we have $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \int _ { u } ^ { v } \varphi ( x ) \mathrm { d } x$$ then that $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \right\} \right) = 1 - \Phi ( u )$$
Q32 Central limit theorem View
Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
Show that there exists $x _ { 0 } \geqslant 1$ such that we have $$\forall x \geqslant x _ { 0 } , \quad \exists n _ { x } \in \mathbb { N } , \quad \forall n \geqslant n _ { x } , \quad x ^ { 2 } \mathbb { P } \left( \left\{ \left| S _ { n } \right| \geqslant x \sqrt { n } \right\} \right) \leqslant \varepsilon .$$
Q33 Central limit theorem View
Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
For $x _ { 0 }$ and $x$ as in the previous question, we fix $N \geqslant \frac { n _ { x } } { \varepsilon }$ and we choose $n \geqslant N$. Show that then $$x ^ { 2 } \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| S _ { p } \right| \geqslant 3 x \sqrt { n } \right\} \right) \leqslant 3 \varepsilon$$