Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 8

For a nonnegative integer-valued random variable, its probability generating function isThe series converges absolutely at least for $|s|\le 1$, with $0^0=1$ in the $k=0$ term; it may converge on a larger domain.

This is the geometric distribution on $\{0,1,2,\dots \}$ with success parameter $1/2$. Its probability generating function is a geometric series:

The ancestor contributes one individual. Given its offspring count $N$, the descendant family rooted at each child is an independent branching process with the same offspring distribution as the original one. Its total progeny therefore has the same distribution as $Z$. The independent reproduction choices let these descendant total progeny variables be represented by independent copies $Z_1,Z_2,\dots$, independent of $N$. Hence, on this family-tree construction,where the sum is zero when $N=0$.

For $0\le s\le 1$, condition on $N=k$. Independence of the descendant total progeny variables givesincluding the empty product $1$ when $k=0$. Taking expectation over $N$ proves the probability generating function of an independent random sum formulaThe same identity holds for complex $|s|<1$ by absolute convergence.

The extra ancestor multiplies the probability generating function by $s$. The preceding probability generating function of an independent random sum formula therefore yieldsFor $0\le s<1$ the two roots are $1\pm \sqrt{1-s}$. A probability generating function lies in $[0,1]$ there, so the plus root is impossible. Also $Z\ge 1$ gives $G_Z(0)=0$. Consequently the total progeny with critical geometric offspring hasAt $s=1$ the value is $1$, consistent with almost-sure finiteness. For complex $|s|<1$, the same expression uses the analytic square root that equals $1$ at $s=0$.