= Solution
{c}
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 $N$.
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