Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 7 / b / iii / Solution

Set $D(s)=4-2s-s^2$, so $D(s)G(s)=s^2$. Differentiate once and evaluate at $s=1$, using $D(1)=1$, $D^{\prime }(1)=-4$, $G(1)=1$:A second differentiation gives $D^{\prime \prime }G+2D^{\prime }{G}^{\prime }+D{G}^{\prime \prime }=2$, henceThe derivatives of a probability generating function give $G^{\prime }(1)=\mathbb{E}[F]$ and $G^{\prime \prime }(1)=\mathbb{E}[F(F-1)]$. Consequently $\mathrm{Var}(F)=52+6-6^2=22$. The earlier geometric tail bound from a uniform escape probability ensures the needed moments are finite.