= Expected generation size in a Galton-Watson process
{title2=$\mathbb E[Z_n]=\mu^n$}
For a <Galton-Watson process> starting with one ancestor and finite offspring <mean> $\mu$, the <expected value> of the generation size is $\mathbb E[Z_n]=\mu^n$. At $n=0$ it is one, including when $\mu=0$; zero offspring <mean> forces every later generation to be empty. With $m$ initial ancestors, the <expected value> is $m\mu^n$.
For $p_0=p_1=1/4$, $p_2=1/2$, the offspring <mean> is $5/4$. The exact calculations used in the generation <probability mass function> figure under <Galton-Watson process> give:
|| Generation $n$
|| Possible sizes
|| Extinction by $n$: $q_n$
|| <Expected value> $\mathbb E[Z_n]$
| 0
| 1
| 0
| 1
| 1
| 0–2
| 0.2500000000
| 1.25
| 2
| 0–4
| 0.3437500000
| 1.5625
| 3
| 0–8
| 0.3950195313
| 1.953125
| 4
| 0–16
| 0.4267750978
| 2.44140625
| 5
| 0–32
| 0.4477622665
| 3.0517578125
| 6
| 0–64
| 0.4621860903
| 3.814697265625
Nearly half the families are already empty at generation six, yet the <expected value> exceeds three. Since empty families contribute zero,
$$
\mathbb E[Z_n\mid Z_n>0]=\frac{\mu^n}{1-q_n}
$$
whenever $q_n<1$. At generation six this <conditional expectation> is about $7.092969$. Conditioning on being nonempty now differs from conditioning on eventual survival: some presently nonempty families will still die out. The eventual <branching extinction probability> is $1/2$, even as the unconditional <expected value> grows without bound.
The <expected value> is useful for computing average population counts and, in the subcritical case, the <expected value> of the <total progeny of a branching process>. Its simple formula comes from one local calculation: <linearity of expectation> and the <independent and identically distributed> offspring counts give
$$
\mathbb E[Z_{n+1}\mid Z_n]=\mu Z_n,\qquad \mathbb E[Z_{n+1}]=\mu\mathbb E[Z_n].
$$
The <tower property of conditional expectation> gives the second equality; starting from $\mathbb E[Z_0]=1$ gives the formula. It describes an average over the generation <probability distribution>, not deterministic growth of an individual family. 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}
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