# Math Help - Extinction Theorem

1. ## Extinction Theorem

Is anyone familiar with this theorem:

If E(X) > 1 then the extinction probability $\pi$ is the unique root of the equation $\pi = f(\pi)$, which lies strictly between 0 and 1.

If E(X) < 1, then $\pi = 1$

Graph is below:

I cannot seem to find any link between the expectations, and the curve f(x) and y=x in the graph. And what is the meaning of $\pi_1, \pi_2$?
2. It looks as though π is meant to be the limit of the increasing sequence $\pi_1,\,\pi_2,\,\pi_3,\ldots$ given by $\pi_1=f(0),\;\pi_{n+1}=f(\pi_n)\;\;(n\geqslant1)$. Needless to say, π here does not mean the number 3.14159... .