= Mean and variance of a Galton-Watson generation
{title2=$\mathbb E[X_n]=\mu^n,\quad v_{n+1}=\mu^2v_n+\sigma^2\mu^n$}
= Branching-process generation moments
{synonym}
With one initial ancestor and independent offspring counts of finite mean and <variance>, conditioning on the current generation gives the displayed recurrences. For $\mu\ne1$ the <variance> is $\sigma^2\mu^{n-1}(\mu^n-1)/(\mu-1)$ for positive generation index. At $\mu=1$ it is $n\sigma^2$. At zero offspring mean the distribution is concentrated at zero and later generations vanish. These formulas are for the <Galton-Watson process>, not arbitrary dependent reproduction.
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