= Binomial branching process conditioned on extinction
{title2=$\widetilde p=p\eta/(1-p+p\eta)$}
For offspring $\operatorname{Bin}(k,p)$ with $0<p<1$ and $kp>1$, let $\eta\in(0,1)$ be the <branching extinction probability>. Conditioning on <extinction> changes the offspring law to $\operatorname{Bin}(k,\widetilde p)$ with the displayed parameter. Indeed, the <probability> of $j$ children is multiplied by $\eta^{j-1}$, so the new <probability generating function> is $f(\eta s)/\eta$. Since $f(s)=(1-p+ps)^k$ and $f(\eta)=\eta$, this equals $((1-p+p\eta s)/(1-p+p\eta))^k$. Its <mean> is $k\widetilde p=f'(\eta)<1$, because <strict convexity> of $f(s)-s$ and its zeros at $\eta$ and one force a negative <derivative> at the smaller zero. The conditioned process therefore has an exponentially decaying total-progeny tail. Unconditionally that tail instead converges to the survival mass $1-\eta$, with an exponentially small correction. When <extinction> has <probability> zero, conditioning on it does not define this law.
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