grandes-ecoles 2015 QIV.C

grandes-ecoles · France · centrale-maths1__psi Probability Generating Functions Branching process and extinction probability via PGF iteration

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.

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.

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