Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / b / i / Solution

Take independent Bernoulli random variables $X_i$ of success probability $r$, independently of $N$. On the event $N=n$, the random sum ${\sum }_{i=1}^N{X}_i$ has the binomial distribution with parameters $(n,r)$, including the empty sum zero at $n=0$. It therefore has exactly the specified joint distribution with $N$.

Thus one may realize $\boxed{Z=\sum _{i=1}^N{X}_i}$ on a suitable probability space. This is a distributional construction; it does not require a pre-existing $Z$ to be independent of $N$.