Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 1 down vote accepted

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.