= Branching process conditioned on extinction
{title2=$f_q(s)=f(qs)/q$}
Let $q\in(0,1]$ be the <branching extinction probability> of a <Galton-Watson process> with offspring <probability generating function> $f$. Conditional on extinction, the probability of $j$ children is multiplied by $q^j/q$: every child's descendants must also become extinct. Hence the new <probability generating function> is $f_q$ and its mean is $f'(q)$. If $f'(q)<1$, the <total progeny of a branching process> has conditional <expected value> $1/(1-f'(q))$, and unconditionally $\mathbb E[T;T<\infty]=q/(1-f'(q))$. The <Markov inequality> gives a finite-total-progeny tail bound without discarding the survival mass.
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