Other offspring laws
Section of Galton–Watson branching, part of Stochastic processes
Other offspring laws
| Type | Section |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
If \xi\sim\mathrm{Poisson}(\lambda), the Poisson distribution has generating function f(s)=e^{\lambda(s-1)}. The mean is \lambda. The extinction probability is the smallest solution of q=e^{\lambda(q-1)}. There is no elementary closed form in general. The iteration q_{n+1}=f(q_n) with q_0=0 is still the probability of extinction by generation n. For \lambda\le 1 the limit is 1. For \lambda>1 the limit is strictly less than 1.
For a geometric offspring law supported on \{0,1,2,\ldots\},
\mathbb{P}(\xi=k)=(1-r)r^k, \qquad k\ge 0,\quad 0<r<1,
the generating function is f(s)=(1-r)/(1-rs) and the mean is m=r/(1-r). This is not the geometric law on the common distributions sheet. That law is the number of trials until the first success, supported on \{1,2,\ldots\}. The offspring law here counts children and includes 0.
The fixed-point equation q=(1-r)/(1-rq) rearranges to rq^2-q+(1-r)=0. One root is q=1. Factoring out (1-q) leaves the second root q=(1-r)/r. If r>1/2, then m>1 and (1-r)/r<1, so q=(1-r)/r. If r\le 1/2, then m\le 1 and the admissible extinction probability is 1. The threshold is the same mean condition m=1. The number q comes from the whole generating function.
Used in
See also
Offspring probability generating function · Extinction criterion · Galton–Watson process · Mean and variance by generation · Extinction probability · Subcritical decay, and what the model leaves out · Total family size · Generations and dependence