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 $E(T)$.
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 $E(T)$.