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 / b / i

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 $N=0$.