Offspring probability generating function
Definition taught in Galton–Watson branching, part of Stochastic processes
Offspring probability generating function
| Type | Definition |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
The offspring probability generating function is
f(s)=\mathbb{E}(s^{\xi})=\sum_{k\ge 0}p_k s^k, \qquad 0\le s\le 1.
It satisfies f(1)=1. If the mean is finite, f'(1-)=m. If \mathbb{E}[\xi(\xi-1)] is finite, f''(1-)=\mathbb{E}[\xi(\xi-1)].
Used in
- Taught in Galton–Watson branching, where the derivation and examples are.
- All 3 questions on Galton–Watson branching