Consider the following modification of a branching process:
A mature individual produces children according to the generating function $g(t)$. However, an individual becomes mature with probability $\alpha$ and dies before maturity with probability $1-\alpha$. We start with one immature individual.
(a) Find the generating function of the number of individuals in the first two generations.
(b) Suppose that the offspring distribution is geometric with parameter $p$. Determine the extinction probability.
Is my working correct for both?
(a) The number of individuals in the first generation is $\alpha g(t)$ if the pioneer individual matures and 0 if it does not. The number of individuals in second generation would be $\alpha^2g(t)^2$.
(b) The number of individuals in the next generation is $\mathsf P(X=r) = rp(1-p)^{r-1}$. The probability of 0 individuals in the next generation = $p$. Let the extinction probability be $K$, then $$K = p + K\mathsf P(\text{1 individual is generated}) + K^2\mathsf P(\text{2 individuals are generated})+\dots$$ $$=p + p(1-p)K + p(1-p)^2K^2 + \dots$$ $$= p + p(1-p)K/(1-(1-p)K)$$ from sum of infinite GP with $r=\frac K{1-p}$. Solving the RHS we get $$K = \frac p{1-(1-p)K}$$ $$(1-p)K^2-(1-p)K-pK+p=0$$ $$((1-p)K-p)(K-1)=0$$ Now $K\ne1$ as mean of geometric distribution is $\frac1p>1$, so $$K=\frac p{1-p}$$