= Poisson branching process
{c}
{title2=$f(s)=e^{\mu(s-1)}$}
A <Galton-Watson process> with a <Poisson distribution> of offspring of mean $\mu$ has this <probability generating function>. For $\mu=1+\varepsilon>1$ its <branching survival probability> $r$ satisfies $-\log(1-r)=(1+\varepsilon)r$, and $2\varepsilon/(1+2\varepsilon)\leq r\leq2\varepsilon$. Indeed $\varepsilon=\sum_{j\geq1}r^j/(j+1)$ lies between $r/2$ and $r/[2(1-r)]$. These exact inequalities belong to the <Poisson branching process>, not to every finite <binomial branching process>.
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