Probability Generating Functions

Branching processes, extinction probabilities, and offspring distributions

Offspring Distribution

1.20

PGF and Fixed Point (Extinction)

Branching Process Simulation

6
50
G(s) = Σ pₙ sⁿ = E[s^X]  |  Extinction: q = G(q), q = smallest fixed pt in [0,1]

A probability generating function G(s) = Σ pₙ sⁿ encodes the offspring distribution. The mean is G'(1) = μ. The extinction probability q is the smallest fixed point of q = G(q). If μ ≤ 1: q=1 (certain extinction). If μ > 1: q < 1 (positive survival probability 1-q). This is the Galton-Watson theorem (1874), originally used to study survival of family surnames. The PGF plot shows G(s) vs the diagonal y=s; their intersection gives the extinction probability.