Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 8 / b / iii

Differentiate the probability generating function equation:At $s=1$, $G_N(1)=1$, givingThe derivative is finite because $p<1/2$ and the explicit square root has no singularity at $s=1$. Alternatively, the expected population in generation $t$ is $(2p{)}^t$, so the expected number of splits in that generation is $p(2p{)}^t$. Summing this nonnegative geometric series gives the same answer without initially assuming a finite expected value.