Generation distribution for Poisson offspring

Question

If a certain animal has $N$ offspring, with $N$ Poisson distributed, what is the distribution of $Z$, the surviving offspring in the next generation (each offspring has probability p of surviving)? I found online theory about Poisson branching processes and there is a probability generating function mentioned there, but I want to know what the exact distribution is. For uniform distributed offspring it is easy to just multiply $p$ with the mean, but here I have some trouble linking the Poisson distribution to the probability of survival $p$.

Answer 1

Do you mean by $Z$ the number of surviving offsprings?

Then $[Z \vert N = n ] \sim$ Bin$(n,p)$ (in words: given $N=n$ the random variable $Z$ has distribution Bin$(n,p)$). This is because for every offspring you are flipping a coin with succes-probability $p$, where the number of offsprings is Poisson distributed.

Hence you will find the answer to your question in Binomial random variable with number of trials being a Poisson random variable.