Mean and variance by generation
Section of Galton–Watson branching, part of Stochastic processes
Mean and variance by generation
| Type | Section |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
Conditional on Z_n, the next generation is a sum of Z_n independent copies of \xi, so its conditional mean is m Z_n and its conditional variance is \sigma^2 Z_n. The tower property and the law of total variance from Stochastic processes therefore give \mathbb{E} Z_{n+1}=m\,\mathbb{E} Z_n and
v_{n+1}=m^2 v_n+\sigma^2 m^n,
where v_n=\operatorname{Var}(Z_n) and v_0=0. The mean iteration starts at \mathbb{E} Z_0=1, so
\mathbb{E} Z_n=m^n.
Unfold the recurrence. The step from n-1 contributes \sigma^2 m^{n-1}, the step from n-2 is multiplied by m^2 once it has entered, and so on back to v_0=0:
v_n=\sum_{j=0}^{n-1} m^{2j}\,\sigma^2 m^{n-1-j}=\sigma^2 m^{n-1}\sum_{j=0}^{n-1} m^j.
For m\neq 1 the geometric sum is (m^n-1)/(m-1), so
\operatorname{Var}(Z_n)=\sigma^2 m^{n-1}\frac{m^n-1}{m-1}\qquad(m\neq 1),
and \operatorname{Var}(Z_n)=n\sigma^2 when m=1. The formula stays non-negative when m<1, because m^n-1 and m-1 have the same sign. The critical case is the limit of the same expression: (m^n-1)/(m-1)\to n as m\to 1, so \sigma^2 m^{n-1} times that ratio tends to n\sigma^2.
One check at n=1: the variance should be \sigma^2. The formula gives \sigma^2 m^{0}(m-1)/(m-1)=\sigma^2.
Used in
See also
Offspring probability generating function · Extinction criterion · Galton–Watson process · Extinction probability · Subcritical decay, and what the model leaves out · Other offspring laws · Total family size · Generations and dependence