grandes-ecoles 2015 QV.C.1

grandes-ecoles · France · centrale-maths1__psi Discrete Random Variables Integral or Series Representation of Moments

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 $$u_n^{(r)}=\sum_{i=1}^{n-1}u_i u_{n-i}^{(r-1)}$$

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
$$u_n^{(r)}=\sum_{i=1}^{n-1}u_i u_{n-i}^{(r-1)}$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B QI.C QI.D.1 QI.D.2 QI.E.1 QI.E.2 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.C.1 QIII.C.2 QIII.C.3 QIV.A QIV.B QIV.C QIV.D QIV.E QIV.F QIV.G QV.A QV.B.1 QV.B.2 QV.C.1 QV.C.2 QV.D.1 QV.D.2 QV.E QV.F