Branching process conditioned on extinction ($f_q(s)=f(qs)/q$)

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 $f^{\prime }(q)$. If $f^{\prime }(q)<1$, the total progeny of a branching process has conditional expected value $1/(1-f^{\prime }(q))$, and unconditionally $\mathbb{E}[T;T<\infty ]=q/(1-f^{\prime }(q))$. The Markov inequality gives a finite-total-progeny tail bound without discarding the survival mass.