= Offspring distribution of a branching process
{title2=$p_k=\mathbb P(\xi=k),\quad F(s)=\sum_{k\geq0}p_ks^k$}
= Offspring distribution
{synonym}
The offspring <probability distribution> of a <Galton-Watson process> is the law of one individual's number of children: $p_k=\mathbb P(\xi=k)$ for $k\geq0$, with $p_k\geq0$ and $\sum_{k\geq0}p_k=1$. Its <probability generating function> and, when finite, its <expected value> are
$$
F(s)=\mathbb E[s^\xi]=\sum_{k\geq0}p_ks^k,\quad 0\leq s\leq1,\qquad \mu=\mathbb E[\xi]=\sum_{k\geq0}kp_k=F'(1-).
$$
Use $0^0=1$ when evaluating the <probability generating function> at zero: $F(0)=p_0$ and $F(1)=1$.
For $p_0=p_1=1/4$ and $p_2=1/2$, $F(s)=1/4+s/4+s^2/2$, $\mu=5/4$, and the offspring <variance> is $11/16$: $\mathbb E[\xi^2]=9/4$ and $\operatorname{Var}(\xi)=9/4-(5/4)^2$. The generation <probability mass functions> pictured under <Galton-Watson process> come from repeatedly composing this <probability generating function>. They describe the whole generation, whereas $(p_k)$ describes one individual's reproduction. https://www.uni-muenster.de/Stochastik/lehre/WS1011/SpezielleStochastischeProzesse/Ch_1.pdf[University of Münster branching-process notes, Section 1.3, Proposition 1.4]{ref}
The whole <offspring distribution> matters for extinction. Another law, $p_0=3/8$ and $p_2=5/8$, has the same offspring <mean> $5/4$ but <branching extinction probability> $3/5$, since $3/8+5s^2/8=s$ has <roots of a polynomial> $3/5$ and $1$. Consequently the two laws have the same <expected generation size in a Galton-Watson process> at each generation but different survival chances. A <binomial branching process> and a <Poisson branching process> likewise specify different <offspring distributions>; a common <mean> does not make their <probability generating functions> identical.
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