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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

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1 Answer 1

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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, $R(s)=sG(R(s))$.

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how do you know you can split it like this? i.e we say originally we have just 1 cell, and that cell producess 1,2,3,,.... which is K? –  Rosie Jun 4 '12 at 13:28
    
Well, this is what the stochastic dynamics means. If the original cell produces $Z=1$ cell, the total number of cells is $K=1+K_1$ ($1$ for the original cell $c$, and $K_1$ for its unique descendant cell $c_1$ plus all the cells produced by $c_1$). –  Did Jun 4 '12 at 13:42
    
why isnt it just 2? If the first cell produces 1 cell, why isnt there just 2?! –  Rosie Jun 4 '12 at 14:02
    
Because the cell $c_1$ produced by the first cell $c$ will itself produce some other cells and one counts the total collection of cells when the process ends. –  Did Jun 4 '12 at 14:16
    
hang on, all through my notes I have an expression but it is never explained. So Xn is the number in the nth generation, and Z1.... etc are iid family sizes. Then $\p_{ij} = P[X_{n+1}=j|X_n=i]=P[Z_1+...+Z_i=j|X_0=i]$. So does this mean that the probability of there being j individuals in the n+1th generation, given there are i individuals in the previous (nth) generation is the same as the probability that you begin with i individuals, and each of those produces a number of individuals, (so individual 1 produces Z1.. etc)such that the number produced in total is j? –  Rosie Jun 4 '12 at 14:29

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