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
2015 centrale-maths1__psi

42 maths questions

QI.A.1 Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
Show that the sequence $(u_n)_{n\in\mathbb{N}}$ is increasing, then that it is convergent. We denote its limit by $l$.
QI.A.2 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
Show that the equation $f(x)=x$ admits a smallest solution. In what follows, we denote it by $x_f$.
QI.A.3 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$. We denote by $x_f$ the smallest solution of $f(x)=x$ and by $l$ the limit of $(u_n)$.
Show that $l=x_f$.
QI.B Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We set $m=f'(1)$. We denote by $x_f$ the smallest solution of $f(x)=x$.
We assume $m>1$. Show that $x_f\in[0,1[$.
QI.C Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We set $m=f'(1)$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$. We denote by $x_f$ the smallest solution of $f(x)=x$.
We now assume $m\leqslant 1$. Show that $x_f=1$ and that for all $n\in\mathbb{N}$, $u_n\neq 1$.
QI.D.1 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We set $m=f'(1)$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
In this question, we assume $m=1$. We set, for $n\in\mathbb{N}$, $\varepsilon_n=1-u_n$. Show that $\lim_{n\rightarrow+\infty}\left(\frac{1}{\varepsilon_{n+1}}-\frac{1}{\varepsilon_n}\right)=\frac{f''(1)}{2}$.
QI.D.2 Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We set $m=f'(1)$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
In this question, we assume $m=1$. We set, for $n\in\mathbb{N}$, $\varepsilon_n=1-u_n$. Deduce that, as $n$ tends to infinity, $1-u_n\sim\frac{2}{f''(1)n}$.
One may use Cesaro's lemma: if $(a_n)_{n\in\mathbb{N}}$ is a sequence of real numbers converging to $l$ and if we set, for $n\in\mathbb{N}^*$, $b_n=\frac{1}{n}(a_1+\cdots+a_n)$, then the sequence $(b_n)_{n\geqslant 1}$ converges to $l$.
QI.E.1 Sequences and series, recurrence and convergence Series convergence and power series analysis View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We set $m=f'(1)$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
We now assume $m<1$ and we set again, for $n\in\mathbb{N}$, $\varepsilon_n=1-u_n$.
Show that the series with general term $\varepsilon_n$ is absolutely convergent and deduce the convergence of the series with general term $\ln\left(\frac{m^{-(n+1)}\varepsilon_{n+1}}{m^{-n}\varepsilon_n}\right)$.
QI.E.2 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View
We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We set $m=f'(1)$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
We now assume $m<1$ and we set again, for $n\in\mathbb{N}$, $\varepsilon_n=1-u_n$.
Deduce that there exists $c>0$ such that, as $n$ tends to infinity, $1-u_n\sim cm^n$.
QII.A.1 Probability Generating Functions PGF of sum of independent variables View
Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
Show that, if $X$ and $Y$ are two independent random variables taking values in $\mathbb{N}$, then $G_{X+Y}=G_X G_Y$.
QII.A.2 Probability Generating Functions PGF of sum of independent variables View
Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
By admitting that, for all $k\in\mathbb{N}$, $S_k$ is independent of $X_{k+1}$, prove that, for all $k\in\mathbb{N}$, $G_{S_k}=(G_X)^k$.
QII.A.3 Probability Generating Functions Compound or random-sum PGF View
Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
By admitting that, for all $n\in\mathbb{N}$, $T$ and $S_n$ are independent, show that $$\forall t\in\left[0,1\left[,\forall K\in\mathbb{N}\quad G_S(t)=\sum_{k=0}^K P(T=k)\left(G_X(t)\right)^k+\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$$
QII.A.4 Probability Generating Functions Bounding probabilities or tail estimates via PGF View
Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
For $K\in\mathbb{N}$ and $t\in\left[0,1\left[$, we set $R_K=\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$.
Show that $0\leqslant R_K\leqslant\frac{1}{1-t}\sum_{k=K+1}^\infty P(T=k)$.
QII.A.5 Probability Generating Functions Compound or random-sum PGF View
Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
Conclude that $G_S=G_T\circ G_X$.
QII.B Probability Generating Functions Deriving moments or distribution from a PGF View
Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$. We have shown $G_S=G_T\circ G_X$.
Deduce that, if $T$ and the $X_n$ have finite expectation, then so does $S$ and $E(S)=E(T)E(X_1)$.
QII.C.1 Poisson distribution View
During a laying, an insect lays a random number of eggs following a Poisson distribution with parameter $\lambda>0$. Then, the probability that a given egg becomes a new insect is $\alpha\in]0,1[$.
Recall the generating function of a random variable following a Poisson distribution with parameter $\lambda$.
QII.C.2 Poisson distribution View
During a laying, an insect lays a random number of eggs following a Poisson distribution with parameter $\lambda>0$. Then, the probability that a given egg becomes a new insect is $\alpha\in]0,1[$.
Using the composition relation $G_S=G_T\circ G_X$, determine the distribution of the number of insects resulting from the laying.
QIII.A.1 Probability Generating Functions Branching process and extinction probability via PGF iteration View
Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with random variables $(X_{n,i})_{(n,i)\in\mathbb{N}\times\mathbb{N}^*}$ independent and all following distribution $\mu$, with $Y_0=1$ and $$\begin{cases} Y_{n+1}(\omega)=0 & \text{if }Y_n(\omega)=0\\ Y_{n+1}(\omega)=\sum_{i=1}^{Y_n(\omega)}X_{n,i}(\omega) & \text{if }Y_n(\omega)\neq 0\end{cases}$$ We denote by $f$ the generating function of $\mu$ and, for $n\in\mathbb{N}$, $\varphi_n$ the generating function of $Y_n$. We have $\varphi_0(t)=t$ for $t\in[0,1]$.
Show that, for all $n\in\mathbb{N}$, $\varphi_{n+1}=\varphi_n\circ f$.
QIII.A.2 Probability Generating Functions Deriving moments or distribution from a PGF View
Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We assume that every random variable following distribution $\mu$ has expectation equal to $m$ and a variance.
Express, for $n\in\mathbb{N}$, the expectation of $Y_n$ in terms of $m$ and $n$.
QIII.A.3 Probability Generating Functions Branching process and extinction probability via PGF iteration View
Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We denote by $f$ the generating function of $\mu$ and $\varphi_n$ the generating function of $Y_n$.
a) Verify that the probability of extinction is equal to the limit of the sequence $(\varphi_n(0))_{n\geqslant 0}$.
b) Verify that we can apply the results of Part I to the sequence $(\varphi_n(0))_{n\geqslant 0}$.
QIII.A.4 Probability Generating Functions Branching process and extinction probability via PGF iteration View
Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We denote by $f$ the generating function of $\mu$ and $m$ the expectation of $\mu$.
If $m\leqslant 1$, show that the probability of extinction is equal to 1.
QIII.B.1 Probability Generating Functions Branching process and extinction probability via PGF iteration View
We consider the Galton-Watson process with extinction time $T$ defined by: $$\omega\in\Omega\quad\begin{cases}T(\omega)=\min\{n\in\mathbb{N}\mid Y_n(\omega)=0\} & \text{if there exists }n\in\mathbb{N}\text{ such that }Y_n(\omega)=0\\ T(\omega)=-1 & \text{otherwise}\end{cases}$$
We assume $m<1$. Verify that $T$ has a finite expectation.
QIII.B.2 Probability Generating Functions Branching process and extinction probability via PGF iteration View
We consider the Galton-Watson process with extinction time $T$. We assume $m<1$.
a) Show that, for all integer $n$, $P(Y_n\geqslant 1)\leqslant m^n$.
b) Show that $E(T)=\sum_{n=0}^{+\infty}P(T>n)$.
c) Deduce an upper bound for $E(T)$.
QIII.C.1 Probability Generating Functions Branching process and extinction probability via PGF iteration View
We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We admit that $Z$ is a random variable defined on $\bigcup_{k\in\mathbb{N}}\{Y_k=0\}$.
Show that $Z$ is defined on a set of probability 1.
QIII.C.2 Probability Generating Functions Deriving moments or distribution from a PGF View
We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$.
a) Show that, for all $k\in\mathbb{N}$, $(P(Z_n\leqslant k))_{n\in\mathbb{N}^*}$ is a convergent sequence. Determine its limit.
b) Deduce that, for all $k\in\mathbb{N}$, $(P(Z_n=k))_{n\in\mathbb{N}^*}$ converges to $P(Z=k)$.
c) Show that, for all $s\in\left[0,1\left[$, all $n\in\mathbb{N}^*$ and $K\in\mathbb{N}$, $$\left|G_{Z_n}(s)-G_Z(s)\right|\leqslant\sum_{k=0}^K\left|P(Z_n=k)-P(Z=k)\right|+\frac{s^K}{1-s}$$
d) Deduce that the sequence of functions $(G_{Z_n})$ converges pointwise to $G_Z$ on $[0,1]$.
QIII.C.3 Probability Generating Functions Compound or random-sum PGF View
We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We denote by $f$ the generating function of $\mu$ and $m$ the expectation of $\mu$.
a) Express $G_{Z_1}$ in terms of $f$.
b) We admit that, for all natural integer $n$ greater than or equal to 2 and for all $s\in[0,1]$, $G_{Z_n}(s)=sf(G_{Z_{n-1}}(s))$.
Deduce that, for all $s\in\left[0,1\left[$, $G_Z(s)=sf(G_Z(s))$.
c) Show that $Z$ has finite expectation if and only if $m<1$. Calculate the expectation when this is the case.
QIV.A Probability Generating Functions Explicit computation of a PGF or characteristic function View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Express, for $t\in[0,1]$, $f(t)$ and calculate $m$.
QIV.B Probability Generating Functions Branching process and extinction probability via PGF iteration View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Verify that, for all $t\in\left[0,1\left[$, $\varphi_n(t)\neq 1$.
QIV.C Probability Generating Functions Branching process and extinction probability via PGF iteration View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)\neq 1$ for $t\in[0,1[$, so we can set $a_n(t)=\frac{1}{\varphi_n(t)-1}$.
Show that, for $t\in\left[0,1\left[$, the sequence $(a_n(t))_{n\in\mathbb{N}}$ is arithmetic.
QIV.D Probability Generating Functions Branching process and extinction probability via PGF iteration View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Deduce that, for $t\in\left[0,1\left[$ and $n\in\mathbb{N}$, $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.
QIV.E Probability Generating Functions Inversion or recovery of distribution from generating/characteristic function View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.
Express, for $(n,k)\in\mathbb{N}^2$, $P(Y_n=k)$ in terms of $n$ and $k$.
QIV.F Probability Generating Functions Branching process and extinction probability via PGF iteration View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$. The extinction time $T$ is defined as before.
Express, in terms of $n\in\mathbb{N}^*$, the probability of the event $T>n$.
Does the variable $T$ have a finite expectation?
QIV.G Probability Generating Functions Deriving moments or distribution from a PGF View
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Express, for $s\in\left[0,1\left[$, $G_Z(s)$ in terms of $s$.
Deduce the distribution of $Z$.
QV.A Probability Generating Functions Radius of convergence and analytic properties of PGF View
We assume $m>1$. We study a slightly different problem: $k$ being a fixed strictly positive integer, we assume that there are $k$ individuals in generation 0. We denote by $W_n$ the number of individuals in the $n$-th generation and define $u_n$ as the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the first time at rank $n$: $$u_n=P\left((W_n=k)\cap\left(\bigcap_{i=1}^{n-1}(W_i\neq k)\right)\right)$$ For $n$ and $r$ non-zero natural integers, $u_n^{(r)}$ is the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the $r$-th time at rank $n$.
Verify that the series $\sum_{n\geqslant 1}u_n s^n$ and $\sum_{n\geqslant 1}u_n^{(r)}s^n$ converge when $s\in[-1,1]$.
QV.B.1 Probability Generating Functions Branching process and extinction probability via PGF iteration View
We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that $P(W_1>k)>0$.
QV.B.2 Probability Generating Functions Branching process and extinction probability via PGF iteration View
We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ does not take the value $k$ is non-zero; we denote this probability by $u$.
One may study separately the cases $p_0=0$ and $p_0>0$.
QV.C.1 Discrete Random Variables Integral or Series Representation of Moments View
We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$. We define $u_n$ and $u_n^{(r)}$ as above.
Let $n\in\mathbb{N}^*$ and $r$ a natural integer greater than or equal to 2. Show the relation $$u_n^{(r)}=\sum_{i=1}^{n-1}u_i u_{n-i}^{(r-1)}$$
QV.C.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0. We define $u_n$, $u_n^{(r)}$, $U(s)=\sum_{n=1}^{+\infty}u_n s^n$ and $U_r(s)=\sum_{n=1}^{+\infty}u_n^{(r)}s^n$ for $s\in[-1,1]$.
Deduce that, for every strictly positive integer $r$, $U_r=U^r$ ($U^r$ denotes $U\times U\times\cdots\times U$ $r$ times).
QV.D.1 Probability Definitions Almost Sure Convergence and Measure-Theoretic Probability View
We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$. We define $u_n$, $u_n^{(r)}$, $U(s)$ and $U_r(s)$ as above, and $u$ is the probability that $(W_n)$ does not take the value $k$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ infinitely many times is zero.
QV.D.2 Probability Definitions Almost Sure Convergence and Measure-Theoretic Probability View
We assume $m>1$. We study the Galton-Watson process with $Y_n$ the number of individuals in generation $n$ (starting from 1 individual).
Show that the probability that the sequence $(Y_n)_{n\in\mathbb{N}^*}$ takes any fixed value $k$ infinitely many times is zero.
QV.E Probability Definitions Almost Sure Convergence and Measure-Theoretic Probability View
Let $(A_n)_{n\in\mathbb{N}}$ be a sequence of events all with probability 1.
Show that $P\left(\bigcup_{n\in\mathbb{N}}\overline{A_n}\right)=0$. What can be deduced for $P\left(\bigcap_{n\in\mathbb{N}}A_n\right)$?
QV.F Probability Definitions Almost Sure Convergence and Measure-Theoretic Probability View
We assume $m>1$. Let $\alpha$ be the probability of extinction and $\beta$ be the probability that the sequence $(Y_n)$ diverges to infinity.
Show that $\alpha+\beta=1$.