Generating functions and tumour cells I have got G(s) = p+ $\ rs^2$ a p.g.f for a family size. Let K be the total number of tumour cells produced from a single original tumour cell Let R(s) = P[K=0] + sP[K=1]+.... be the p.g.f of this number Let the number of immediate descendants of the original cell be Z then K= 1+ $\ K_1+...+K_Z $ where$\ K_1...K_Z$ are independent random total numbers of cells produced by each of the immediate descendants then R(s) = sG(R(s)) I can't figure the last statement out and therefore I can't follow the rest of the example which follows, can anyone explain this? thanks

For every $n\geqslant0$ and every fixed $s$, $\mathrm E(s^K\mid Z=n)=s\mathrm E(s^{K_1})\cdots \mathrm E(s^{K_n})=sR(s)^n$.

Hence, $\mathrm E(s^K)=\mathrm E(\mathrm E(s^K\mid Z))=\mathrm E(sR(s)^Z)=s\mathrm E(R(s)^Z)$. That is, $R(s)=sG(R(s))$.