Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 11F (Probability)

Conditional on $X_n=m$, the next generation is the sum of $m$ independent offspring counts, each with probability generating function $F_1(s)$. HenceThis also holds for $m=0$, when the empty sum is zero and the empty product is one. The law of total expectation givesFor complex $|s|\le 1$, the expectations are absolutely bounded because $|F_1(s)|\le 1$. This proves the branching-process generating-function iteration, with $F_0(s)=s$.

The eventual extinction probability is the smallest solution $\boxed{q\in [0,1]\text{ of }{F}_1(q)=q}$. This is the Galton-Watson extinction fixed point result for one initial individual. Equivalently,since having an empty generation is an absorbing event and the extinction-by-generation events increase. In particular extinction is certain when the smallest fixed point is one. With $p_0>0$, an offspring mean at most one implies this, whereas a mean greater than one gives a fixed point below one. The fixed-point statement itself requires no finite-mean assumption.

Summing the geometric series for the offspring probability generating function givesThe fixed-point equation $s=1/(2-s)$ is $(s-1{)}^2=0$, whose only root in $[0,1]$ is one. The Galton-Watson extinction fixed point result therefore proves extinction is certain.

For the explicit iterates, the proposed formula at $n=1$ is $1/(2-s)$, so the base case holds. If it holds at $n$, part (a) givesThis is exactly the same formula with $n$ replaced by $n+1$. By induction,Evaluating the probability generating function at zero selects the mass of an empty generation:These probabilities increase to one, also confirming the extinction conclusion directly.