grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2019 centrale-maths1__psi

38 maths questions

Q1 Differential equations First-Order Linear DE: General Solution View
Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
Specify the domain of definition $D$ of $f_{\alpha}$. Justify that $f_{\alpha}$ is of class $C^{1}$ on $D$ and give a first-order linear differential equation satisfied by $f_{\alpha}$ on $D$.
Q2 Taylor series First-Order Linear DE: General Solution View
Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
State Cauchy's theorem for a scalar first-order linear differential equation and prove that, for all $x \in ]-1,1[$, $$f_{\alpha}(x) = \sum_{n=0}^{+\infty} L_{n}(\alpha) \frac{x^{n}}{n!}.$$
Q3 Generalised Binomial Theorem and Partial Fractions Properties and Manipulation of Power Series or Formal Series View
Recall the definition of the Cauchy product of two power series and state the theorem relating to it.
Q4 Sequences and Series Functional Equations and Identities via Series View
Using the Cauchy product of power series, deduce that, for all integers $n$ and all real numbers $\alpha$ and $\beta$, $$L_{n}(\alpha + \beta) = \sum_{k=0}^{n} \binom{n}{k} L_{k}(\alpha) L_{n-k}(\beta).$$
Q5 Sequences and Series Evaluation of a Finite or Infinite Sum View
For $x \in ]-1,1[$, give the value of the sum of the power series $\sum_{p=1}^{+\infty} x^{p}$ as well as that of its derivative.
Q6 Sequences and Series Prove existence or uniqueness of an object by induction View
Prove by induction that, for all integers $n \in \mathbb{N}^{*}$, there exists a unique polynomial $R_{n} \in \mathbb{R}_{n}[X]$ such that, for all $x \in ]-1,1[$, $$\sum_{p=1}^{+\infty} p^{n} x^{p} = \frac{R_{n}(x)}{(1-x)^{n+1}}$$
Q7 Discrete Probability Distributions Explicit computation of a PGF or characteristic function View
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$, where $g_{n}(t) = \sum_{k=0}^{+\infty} P(X_{n} = k) t^{k}$.
Determine the distributions of $X_{1}, X_{2}$ and $X_{3}$ and then the functions $g_{1}, g_{2}$ and $g_{3}$.
Q8 Discrete Probability Distributions Recurrence Relations and Sequences Involving Probabilities View
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.
Let $n$ and $k$ be two integers greater than or equal to 1. Establish that $$P(X_{n} = k) = \frac{k-1}{n+1} P(X_{n-1} = k-1) + \frac{n+1-k}{n+1} P(X_{n-1} = k).$$
Q9 Discrete Probability Distributions Recursive or recurrence relation via PGF coefficients View
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.
Using the recurrence relation from question 8, deduce that, for all integers $n$ greater than or equal to 1 and all real $t$, $$g_{n}(t) = \frac{t^{2} - t}{n+1} g_{n-1}^{\prime}(t) + g_{n-1}(t)$$
Q10 Discrete Probability Distributions Deriving moments or distribution from a PGF View
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.
Prove that, for all integers $n \in \mathbb{N}^{*}$ and all real $t$, $$g_{n}(t) = \frac{1}{n+1} \sum_{k=1}^{n+1} t^{k}$$
Q11 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.
Identify the distribution of $X_{n}$ and give its expectation.
Q12 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn.
By listing all possible outcomes, give the distribution of $X_{3}$.
Q13 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Verify that $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.
Q14 Permutations & Arrangements Combinatorial Counting (Non-Probability) View
We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, an outcome resulting from $n$ successive draws is modeled by the $n$-tuple indicating the color and number of the balls successively obtained. We denote by $\Omega_{n}$ the set of possible outcomes of these $n$ draws.
By examining the number of balls in the urn just before each draw, justify that, for $n \geqslant 1$, $$\operatorname{card}(\Omega_{n}) = (a_{0} + b_{0}) \times \cdots \times (a_{0} + b_{0} + s(n-1)) = s^{n} L_{n}\left(\frac{a_{0} + b_{0}}{s}\right).$$
Q15 Discrete Probability Distributions Proof of a Probability Identity or Inequality View
We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, $\Omega_{n}$ denotes the set of possible outcomes of $n$ draws, and $X_{n}$ denotes the number of white balls present in the urn after $n$ draws. For $\omega \in \Omega_{n}$, $b(\omega)$ denotes the number of white balls present in the urn at the end of the $n$ draws modeled by $\omega$.
Show that, for all $n \in \mathbb{N}^{*}$ and all $k \in \mathbb{N}$, $$P(X_{n} = k) = \frac{\operatorname{card}(\{\omega \in \Omega_{n} ; b(\omega) = k\})}{\operatorname{card}(\Omega_{n})}.$$
Q16 Probability Generating Functions Deriving or Identifying a Probability Distribution from a Random Process View
We consider a general balanced urn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$. We denote by $g_{n}$ the generating function of $X_{n}$ (the number of white balls after $n$ draws).
Justify the equalities $$\begin{aligned} & g_{n}(t) = \frac{1}{\operatorname{card}(\Omega_{n})} P_{n}(t, 1) \\ & E(X_{n}) = \frac{1}{\operatorname{card}(\Omega_{n})} \frac{\partial P_{n}}{\partial u}(1,1) \end{aligned}$$
Q17 Differential equations Properties and Manipulation of Power Series or Formal Series View
We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Prove that, for all integers $n$, $$P_{n+1}(u,v) = u^{a+1} v^{b} \frac{\partial P_{n}}{\partial u}(u,v) + u^{c} v^{d+1} \frac{\partial P_{n}}{\partial v}(u,v)$$
Q18 Differential equations Uniform or Pointwise Convergence of Function Series/Sequences View
We consider a general balanced urn. For all real $x, u$ and $v$, we set, provided it exists, $$H(x, u, v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$$ Let $\rho > 0$. We set $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2} = \{(x,u,v) \in \mathbb{R}^{3} ; |x| < \rho, 0 < u < 2, 0 < v < 2\}$.
Justify that, for $\rho$ sufficiently small, the function $H$ is well defined on $D_{\rho}$.
Q19 Differential equations Uniform or Pointwise Convergence of Function Series/Sequences View
We consider a general balanced urn. For all real $x, u$ and $v$, we set $$H(x, u, v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$$ defined on $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$ for $\rho$ sufficiently small.
Justify that $H$ admits a first-order partial derivative with respect to $x$ on the domain $D_{\rho}$, obtained by term-by-term differentiation with respect to $x$ of the expression for $H$.
Q20 Differential equations Series convergence and power series analysis View
We consider a general balanced urn. For all real $x, u$ and $v$, we set $$H(x, u, v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$$ defined on $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$ for $\rho$ sufficiently small.
Prove that $H$ admits a first-order partial derivative with respect to $u$ on the domain $D_{\rho}$, obtained by term-by-term differentiation with respect to $u$ of the expression for $H$.
Q21 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We consider a general balanced urn. For all real $x, u$ and $v$, we set $$H(x, u, v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$$ defined on $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$ for $\rho$ sufficiently small.
Verify that $H(0, u, v) = u^{a_{0}} v^{b_{0}}$ and then that $H$ is a solution on $D_{\rho}$ of the partial differential equation $$\frac{\partial H}{\partial x}(x,u,v) = u^{a+1} v^{b} \frac{\partial H}{\partial u}(x,u,v) + u^{c} v^{d+1} \frac{\partial H}{\partial v}(x,u,v).$$
Q22 Differential equations Series convergence and power series analysis View
In the general model of a Pólya urn, we consider the balanced urn model for which $b = c = 0$, so $a = d$. Each time we draw a ball, we add $a$ balls of its color to the urn. The initial composition is $a_{0}$ white balls and $b_{0}$ black balls. The function $G$ is defined on $U = \{(x,u,v) \in \mathbb{R} \times \mathbb{R}_{+}^{*} \times \mathbb{R}_{+}^{*} ; axu^{a} < 1, axv^{a} < 1\}$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ We use the notation $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$.
Using the preliminary results, prove that there exists $\rho > 0$ such that $D_{\rho} \subset U$ and, for all $(x,u,v) \in D_{\rho}$, $$G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$$ where $Q_{n}$ is a polynomial function of two variables to be specified.
Q23 Differential equations Uniform or Pointwise Convergence of Function Series/Sequences View
In the general model of a Pólya urn ($b = c = 0$, $a = d$), the function $G$ is defined on $U$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ and admits the expansion $G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$ on $D_{\rho}$.
Justify that $G$ admits a first-order partial derivative with respect to $x$ on the domain $D_{\rho}$, obtained by term-by-term differentiation with respect to $x$ of the expression for $G$.
Q24 Integration using inverse trig and hyperbolic functions Uniform or Pointwise Convergence of Function Series/Sequences View
In the general model of a Pólya urn ($b = c = 0$, $a = d$), the function $G$ is defined on $U$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ and admits the expansion $G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$ on $D_{\rho}$.
Prove that $G$ admits a first-order partial derivative with respect to $u$ on the domain $D_{\rho}$, obtained by term-by-term differentiation with respect to $u$ of the expression for $G$.
Q25 Integration using inverse trig and hyperbolic functions Properties and Manipulation of Power Series or Formal Series View
In the general model of a Pólya urn ($b = c = 0$, $a = d$), the function $G$ is defined on $U$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ and admits the expansion $G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$ on $D_{\rho}$. The function $H(x,u,v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$ was defined in part III.
Deduce that, for all integers $n$, $P_{n} = Q_{n}$, and then that $H$ and $G$ coincide on $D_{\rho}$.
Q26 Discrete Probability Distributions Distribution of Transformed or Combined Random Variables View
In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the results established so far (in particular that $H = G$ on $D_{\rho}$), conclude that, for all integers $n$ and for all $k \in \llbracket 0, n \rrbracket$, $$P(X_{n} = a_{0} + ka) = \binom{n}{k} \frac{L_{k}(a_{0}/a) L_{n-k}(b_{0}/a)}{L_{n}(a_{0}/a + b_{0}/a)}$$
Q27 Discrete Probability Distributions Distribution of Transformed or Combined Random Variables View
In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the result of question 26, recover the result of question 10.
Q28 Measures of Location and Spread Expectation and Moment Inequality Proof View
In the general model of a Pólya urn ($b = c = 0$, $a = d$), using the results of questions 16 and 19, determine the expectation of $X_{n}$.
Q29 Sequences and Series Functional Equations and Identities via Series View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have $\operatorname{card}(\Omega_{n}) = n!$. For $0 < u < v$ and $|x|$ sufficiently small, $$H(x,u,v) = \sum_{n=0}^{+\infty} \frac{x^{n}}{n!} \left( \sum_{p=1}^{+\infty} p^{n} (v-u)^{n+1} \left(\frac{u}{v}\right)^{p} \right)$$
Using question 6, justify that, for all integers $n$ and all $u$ and $v$ such that $0 < u < v$, the sum $$\sum_{p=1}^{+\infty} p^{n} (v-u)^{n+1} \left(\frac{u}{v}\right)^{p}$$ is a polynomial function of $u$ and $v$.
Q30 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have for all integers $n$ and all $t \in ]0,1[$, $$g_{n}(t) = \frac{1}{n!} \sum_{p=1}^{+\infty} p^{n} t^{p} (1-t)^{n+1}.$$ Fix an integer $n \geqslant 2$.
Show that $\sum_{p=n+1}^{+\infty} p^{n} t^{p} (1-t)^{n+1} \underset{t \rightarrow 0^{+}}{=} O(t^{n+1})$.
Q31 Sequences and Series Power Series Expansion and Radius of Convergence View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have for all integers $n$ and all $t \in ]0,1[$, $$g_{n}(t) = \frac{1}{n!} \sum_{p=1}^{+\infty} p^{n} t^{p} (1-t)^{n+1}.$$ Fix an integer $n \geqslant 2$.
Using the result of question 30 and by expanding $(1-t)^{n+1}$, determine the Taylor expansion of $g_{n}$ to order $n$ at 0.
Q32 Sequences and Series Deriving or Identifying a Probability Distribution from a Random Process View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), fix an integer $n \geqslant 2$.
Using the Taylor expansion of $g_{n}$ to order $n$ at 0, deduce that, for all $m$ in $\llbracket 1, n \rrbracket$, $$P(X_{n} = m) = \frac{1}{n!} \sum_{k=0}^{m-1} (-1)^{k} \binom{n+1}{k} (m-k)^{n}.$$
Q33 Proof Permutation Properties and Enumeration (Abstract) View
Let $k$ be an integer greater than or equal to $1$, $(u_{0}, \ldots, u_{k})$ a finite sequence of integers, and $a$ an integer such that $a > u_{p}$ for all $p \in \llbracket 0, k \rrbracket$. We insert the value $a$ into this sequence immediately after $u_{i}$, with $i \in \llbracket 0, k-1 \rrbracket$, to obtain the sequence $(u_{0}, \ldots, u_{i}, a, u_{i+1}, \ldots, u_{k})$. Compare the number of ascents and descents of the new sequence with respect to the old one. Two cases will be distinguished.
Q34 Proof Permutation Properties and Enumeration (Abstract) View
We denote by $S_{n}$ the set of permutations of $\llbracket 1, n \rrbracket$. We represent an element $\sigma$ of $S_{n}$ by the finite sequence $(\sigma(1), \sigma(2), \ldots, \sigma(n))$ and we call an ascent (respectively descent) of $\sigma$ an ascent of this sequence. For any integer $m$, we denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents.
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.
Determine the elements of $S_{3}$ and calculate among them the number of permutations with $m$ ascents for any integer $m$. Compare the values obtained with the coefficients of $P_{3}(X, 1)$ where $P_{3}$ was expressed in question 13.
Q35 Proof Permutation Properties and Enumeration (Abstract) View
We denote by $S_{n}$ the set of permutations of $\llbracket 1, n \rrbracket$. We represent an element $\sigma$ of $S_{n}$ by the finite sequence $(\sigma(1), \sigma(2), \ldots, \sigma(n))$ and we call an ascent (respectively descent) of $\sigma$ an ascent of this sequence. For any integer $m$, we denote by $A_{n,m}$ the number of elements of $S_{n}$ with $m$ ascents.
Let $n \geqslant 2$. Determine $A_{n,0}$, $A_{n,n-1}$ and $A_{n,m}$ for $m \geqslant n$.
Q36 Proof Permutation Properties and Enumeration (Abstract) View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), an algorithm constructs a permutation of $S_{n}$ from an outcome of $n$ draws as follows:
  • We start with the sequence $(0,1,0)$ after the first draw.
  • For any $k \in \llbracket 2, n \rrbracket$, if on the $k$-th draw we draw a white ball (respectively black ball) numbered $p$, we insert the value $k$ in the sequence in the middle of the $(p+1)$-th ascent (respectively descent).
  • At the end, we remove the two 0's from the list.

Using the algorithm above, construct the permutation of $S_{5}$ associated with the outcome $(B_{0}, B_{0}, N_{1}, N_{0}, B_{2})$.
Q37 Proof Permutation Properties and Enumeration (Abstract) View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), an algorithm constructs a permutation of $S_{n}$ from an outcome of $n$ draws as follows:
  • We start with the sequence $(0,1,0)$ after the first draw.
  • For any $k \in \llbracket 2, n \rrbracket$, if on the $k$-th draw we draw a white ball (respectively black ball) numbered $p$, we insert the value $k$ in the sequence in the middle of the $(p+1)$-th ascent (respectively descent).
  • At the end, we remove the two 0's from the list.

Conversely, let $\sigma$ be the element of $S_{7}$ represented by the sequence $(7,1,3,6,5,4,2)$. Determine an outcome $\omega$ consisting of 7 draws such that $\sigma_{\omega} = \sigma$.
Q38 Proof Deriving or Identifying a Probability Distribution from a Random Process View
In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), an algorithm constructs a permutation of $S_{n}$ from an outcome of $n$ draws. Using question 33, compare, for any outcome, the number of white balls in the final composition of the urn with the number of ascents of the permutation associated with it by the algorithm above.