grandes-ecoles 2015 QIV.F

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 $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$. 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?

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?

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