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

2020 centrale-maths2__pc

40 maths questions

Q1 Moment generating functions Compute MGF or characteristic function for a named distribution View

We assume in this question that $X ( \Omega )$ is a finite set of cardinality $r \in \mathbb { N } ^ { * }$. We denote $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ where the $x _ { i }$ are pairwise distinct, and, for all integer $k \in \llbracket 1 , r \rrbracket , a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all real $t , \phi _ { X } ( t ) = \sum _ { k = 1 } ^ { r } a _ { k } \mathrm { e } ^ { \mathrm { i } t x _ { k } }$.

Q2 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View

We assume in this question that $X ( \Omega )$ is a countable set. We denote $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ where the $x _ { n }$ are pairwise distinct. For all $n \in \mathbb { N }$, we set $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Show that $\phi _ { X }$ is defined on $\mathbb { R }$ and that, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { \mathrm { i } t x _ { n } }$.

Q3 Discrete Random Variables Convergence of Expectations or Moments View

Show that $\phi _ { X }$ is continuous on $\mathbb { R }$.

Q4 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

Let $a$ and $b$ be two real numbers and $Y = a X + b$. For all real $t$, express $\phi _ { Y } ( t )$ in terms of $\phi _ { X } , t , a$ and $b$.

Q5 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

Let $t \in \mathbb { R }$. Give an expression of $\phi _ { X } ( - t )$ in terms of $\phi _ { X } ( t )$. Deduce a necessary and sufficient condition on the image $\phi _ { X } ( \mathbb { R } )$ for the function $\phi _ { X }$ to be even.

Q6 Probability Generating Functions Explicit computation of a PGF or characteristic function View

Let $n \in \mathbb { N } ^ { * }$ and $\left. p \in \right] 0,1 [$. We assume that $X : \Omega \rightarrow \mathbb { R }$ follows a binomial distribution $\mathcal { B } ( n , p )$ and we denote $q = 1 - p$. Show that, for all $t \in \mathbb { R } , \phi _ { X } ( t ) = \left( q + p \mathrm { e } ^ { \mathrm { i } t } \right) ^ { n }$.

Q7 Geometric Distribution View

Let $p \in ] 0,1 [$. What is the characteristic function of a random variable following a geometric distribution with parameter $p$ ?

Q8 Probability Generating Functions Explicit computation of a PGF or characteristic function View

Let $\lambda > 0$. What is the characteristic function of a random variable following a Poisson distribution with parameter $\lambda$ ?

Q9 Moment generating functions Existence and domain of the MGF View

Show that for all $t \in \mathbb { R } , \left| \phi _ { X } ( t ) \right| \leqslant 1$.

Q10 Moment generating functions MGF uniquely determines moments or distribution View

Show that, if there exist $a \in \mathbb { R }$ and $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $X ( \Omega ) \subset a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z }$, then $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$.

Q11 Moment generating functions MGF uniquely determines moments or distribution View

We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Show that there exists $a \in \mathbb { R }$ such that $\sum _ { n = 0 } ^ { + \infty } a _ { n } \exp \left( \mathrm { i } \left( t _ { 0 } x _ { n } - t _ { 0 } a \right) \right) = 1$.

Q12 Moment generating functions MGF uniquely determines moments or distribution View

We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Using the result of Q11, deduce that $\sum _ { n = 0 } ^ { + \infty } a _ { n } \left( 1 - \cos \left( t _ { 0 } x _ { n } - t _ { 0 } a \right) \right) = 0$.

Q13 Moment generating functions MGF uniquely determines moments or distribution View

We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Show that for all $n \in \mathbb { N }$, if $a _ { n } \neq 0$, then $x _ { n } \in a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z }$.

Q14 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Deduce that $\mathbb { P } \left( X \in a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z } \right) = 1$.

Q15 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 1 } ^ { r } \operatorname { sinc } \left( T \left( x _ { n } - m \right) \right) \mathbb { P } \left( X = x _ { n } \right)$.

Q16 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Using the result of Q15, deduce that $V _ { m } ( T ) \xrightarrow [ T \rightarrow + \infty ] { } \mathbb { P } ( X = m )$.

Q17 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is countable and we use the notations of question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$. Show that for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 0 } ^ { + \infty } g _ { n } \left( \frac { 1 } { T } \right)$.

Q18 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$. Show that the function $g _ { n }$ extends to a function $\tilde { g } _ { n }$ defined and continuous on $\mathbb { R } ^ { + }$.

Q19 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$, and $\tilde{g}_n$ denotes its continuous extension to $\mathbb{R}^+$. Show that the function $G = \sum _ { n = 0 } ^ { + \infty } \tilde { g } _ { n }$ is defined and continuous on $\mathbb { R } ^ { + }$.

Q20 Probability Generating Functions Inversion or recovery of distribution from generating/characteristic function View

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is countable and we use the notations of question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Using the results of Q17--Q19, establish that $V _ { m } ( T ) \xrightarrow [ T \rightarrow + \infty ] { } \mathbb { P } ( X = m )$.

Q21 Probability Generating Functions Uniqueness and characterization of distributions via PGF View

Let $X : \Omega \rightarrow \mathbb { R }$ and $Y : \Omega \rightarrow \mathbb { R }$ be two discrete random variables such that $\phi _ { X } = \phi _ { Y }$. Show that, for all $m \in \mathbb { R } , \mathbb { P } ( X = m ) = \mathbb { P } ( Y = m )$, in other words that $X$ and $Y$ have the same distribution.

Q22 Taylor series Prove smoothness or power series expandability of a function View

Q23 Differentiating Transcendental Functions Differentiation under the integral sign with transcendental kernels View

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $N$ be a natural integer and let $F _ { N }$ be the function defined, for all real $x$, by $F _ { N } ( x ) = \int _ { - N } ^ { N } K _ { a , x } ( t ) \mathrm { d } t$. Show that $F _ { N }$ is of class $C ^ { 1 }$ on $\mathbb { R }$ and that, for all real $x , F _ { N } ^ { \prime } ( x ) = N \operatorname { sinc } ( N x )$.

Q24 Integration by Substitution Substitution to Prove an Integral Identity or Equality View

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Show that $\int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t = \int _ { N a } ^ { N b } \operatorname { sinc } ( s ) \mathrm { d } s$.

Q25 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Show that the integral $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s$ is convergent.

Q26 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$. If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Deduce the existence and value of $\lim _ { N \rightarrow + \infty } \int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t$ in the case where $a < b$.

Q27 Moment generating functions MGF uniquely determines moments or distribution View

We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$. If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $X : \Omega \rightarrow \mathbb { R }$ be a random variable such that $X ( \Omega )$ is finite. We assume that the real numbers $a$ and $b$ do not belong to $X ( \Omega )$. Show that $$\frac { 1 } { \pi } \int _ { - N } ^ { N } \phi _ { X } ( - t ) K _ { a , b } ( t ) \mathrm { d } t \xrightarrow [ N \rightarrow + \infty ] { } \mathbb { P } ( a < X < b )$$

Q28 Discrete Random Variables Existence of Expectation or Moments View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Let $j$ be an integer such that $1 \leqslant j \leqslant k$. Show that for all real $x , | x | ^ { j } \leqslant 1 + | x | ^ { k }$ and deduce that $X$ admits a moment of order $j$.

Q29 Moment generating functions Derivative formulas and recursive structure for characteristic functions View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Deduce that $\phi _ { X }$ is of class $C ^ { k }$ on $\mathbb { R }$ and give an expression of the $k$-th derivative of $\phi _ { X }$.

Q30 Probability Generating Functions Deriving moments or distribution from a PGF View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Deduce an expression of $\mathbb { E } \left( X ^ { k } \right)$ in terms of $\phi _ { X } ^ { ( k ) } ( 0 )$.

Q31 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. We denote $f$ the function which to all real $h > 0$ associates $f ( h ) = \frac { 2 \phi _ { X } ( 0 ) - \phi _ { X } ( 2 h ) - \phi _ { X } ( - 2 h ) } { 4 h ^ { 2 } }$. What is the limit of $f$ at 0 ?

Q32 Continuous Probability Distributions and Random Variables Characteristic/Moment Generating Function Derivation View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. We denote $f ( h ) = \frac { 2 \phi _ { X } ( 0 ) - \phi _ { X } ( 2 h ) - \phi _ { X } ( - 2 h ) } { 4 h ^ { 2 } }$ for $h > 0$. Show that for all $h \in \mathbb { R } ^ { * } , f ( h ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \frac { \sin ^ { 2 } \left( h x _ { n } \right) } { h ^ { 2 } }$.

Q33 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. Using the results of Q31 and Q32, deduce that $X$ admits a moment of order 2.

Q34 Discrete Random Variables Existence of Expectation or Moments View

Q35 Discrete Random Variables Existence of Expectation or Moments View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Let $Y : \Omega \rightarrow \mathbb { R }$ be a random variable satisfying $Y ( \Omega ) = X ( \Omega )$ and, for all $n \in \mathbb { N }$, $$\mathbb { P } \left( Y = x _ { n } \right) = \frac { a _ { n } x _ { n } ^ { 2 k } } { \alpha }$$ Show that $\phi _ { Y }$ is of class $C ^ { 2 }$ on $\mathbb { R }$.

Q36 Discrete Random Variables Existence of Expectation or Moments View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Using the result of Q35, deduce that $X$ admits a moment of order $2 k + 2$.

Q37 Discrete Random Variables Existence of Expectation or Moments View

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set. Let $k \in \mathbb { N } ^ { * }$. Deduce from the previous questions that if $\phi _ { X }$ is of class $C ^ { 2 k }$ on $\mathbb { R }$, then $X$ admits a moment of order $2 k$.

Q38 Moment generating functions Power series expansion of the characteristic function View

Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is finite and we use the notation from question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that $\phi _ { X }$ is expandable as a power series on $\mathbb { R }$ and, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( \mathrm { i } t ) ^ { n } } { n ! } \mathbb { E } \left( X ^ { n } \right)$.

Q39 Proof Direct Proof of an Inequality View

Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is countable and we use the notation from question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We also assume that, for all integer $n \in \mathbb { N } , X$ admits a moment of order $n$ and that there exists a real $R > 0$ such that $$\mathbb { E } \left( | X | ^ { n } \right) = O \left( \frac { n ^ { n } } { R ^ { n } } \right) \quad \text { when } n \rightarrow + \infty$$ Show that for all $n \in \mathbb { N }$ and all $y \in \mathbb { R } , \left| \mathrm { e } ^ { \mathrm { i } y } - \sum _ { k = 0 } ^ { n } \frac { ( \mathrm { i } y ) ^ { k } } { k ! } \right| \leqslant \frac { | y | ^ { n + 1 } } { ( n + 1 ) ! }$.

Q40 Moment generating functions Power series expansion of the characteristic function View

Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is countable and we use the notation from question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We also assume that, for all integer $n \in \mathbb { N } , X$ admits a moment of order $n$ and that there exists a real $R > 0$ such that $$\mathbb { E } \left( | X | ^ { n } \right) = O \left( \frac { n ^ { n } } { R ^ { n } } \right) \quad \text { when } n \rightarrow + \infty$$ Using the result of Q39, deduce that for all real $t \in \left[ - \frac { R } { \mathrm { e } } , \frac { R } { \mathrm { e } } \right]$, $$\phi _ { X } ( t ) = \sum _ { k = 0 } ^ { + \infty } \frac { ( \mathrm { i } t ) ^ { k } } { k ! } \mathbb { E } \left( X ^ { k } \right)$$