Editing Galton–Watson process (section)

==Extinction criterion for Galton–Watson process==
In the non-trivial case, the probability of final extinction is equal to 1 if ''E''{''ξ''<sub>1</sub>}&nbsp;≤&nbsp;1 and strictly less than 1 if ''E''{''ξ''<sub>1</sub>}&nbsp;>&nbsp;1.

The process can be treated analytically using the method of [[probability generating function]]s.

If the number of children ''ξ <sub>j</sub>'' at each node follows a [[Poisson distribution]] with parameter λ, a particularly simple recurrence can be found for the total extinction probability&nbsp;''x<sub>n</sub>'' for a process starting with a single individual at time ''n''&nbsp;=&nbsp;0:

:<math>x_{n+1} = e^{\lambda (x_n - 1)},\, </math>

giving the above curves.