Past exam of the theoretical physics major option course of the University of Oxford / 2017 / 3 / e

The rates in part d turn the generating-function equation intoFor the distribution of a critical linear birth-death population, the method of characteristics gives $\dot{z}=-\lambda (z-1{)}^2$ and $\dot{F}=0$. Setting $w=(1-z{)}^{-1}$ gives $\dot{w}=-\lambda$, so the initial characteristic coordinate is $w_0=w+\lambda t$. HenceIt has the correct initial value and $F(1,t)=1$. With $a=\lambda t$, rewrite the denominator as $(1+a)[1-az/(1+a)]$ and expand its geometric series. The probabilities areAt $t=0$ this is understood by continuity: $P_1=1$ and all other probabilities vanish. Summing the positive-state probabilities gives $1/(1+a)$, which together with $P_0$ verifies normalization directly.

There is eventual extinction with probability one, because $P_0(t)\to 1$. Nevertheless $⟨n⟩=F_z(1,t)=1$ at every finite time, while $\mathrm{Var}(n)=2\lambda t$. There is no contradiction: the increasingly rare surviving populations have conditional mean $⟨n\mid n>0⟩=1+\lambda t$. This is the distinction between a constant ensemble mean and survival of a typical realization.