Assuming we have a population with a poisson distribution $p_m=e^{-\lambda} \frac{\lambda^m}{m!}$ I want to find the "survival function f" in relation to $\lambda$. My Approach:
Let P(S) be the survival probability, and P(D) the "Die out" probability. I know that P(S)+P(D)=1, additionaly i know that P(D)=1 iff R $\leq$ 1 so for $\lambda \in [0,1]$ f is 0. Now let g be the generating function of the poisson distribution $g(s)=e^{\lambda (s-1)}$, $P_n$ the size of the population in the nth generation. Then i define $x_n:=P(\sum\nolimits_{n\geq0} P_n=0)$, so that $x_n$ is increasing and $\lim\limits_{n \rightarrow \infty}{x_n}$=:x is a fixpoint of g. Know i dont know how to continue. Thanks in advance