A critical offspring law
For the offspring law \mathbb{P}(\xi=0)=2/5, \mathbb{P}(\xi=1)=1/5, \mathbb{P}(\xi=2)=2/5, find the mean, the probability generating function, and the extinction probability. Explain the result.
Worked answer (click to show)
The mean is
m=0\cdot\frac{2}{5}+1\cdot\frac{1}{5}+2\cdot\frac{2}{5}=1.
The probability generating function is f(s)=2/5+s/5+2s^2/5. The equation f(s)=s is 2s^2-4s+2=0, or (s-1)^2=0. The only root in [0,1] is 1, so the extinction probability is 1. The offspring law is non-degenerate and p_0=2/5>0, so the critical case of the extinction criterion applies.