A population contains \(X_n\) individuals at time \(n=0,1,2,\dots\) . Suppose that \(X_0\) is distributed as \(\mathrm{Poisson}(\mu)\). Between time \(n\) and \(n+1\) each of the \(X_n\) individuals dies with probability \(p\) independent of the others. The population at time \(n+1\) is comprised of the survivors together with a random number of new immigrants who arrive independently in numbers distributed according to \(\mathrm{Poisson}(\mu)\).

- What is the distribution of \(X_n\) ?
- What happens to this distribution as \(n \rightarrow \infty\) ? Your answer should depended on \(p\) and \(\mu\). In particular, what is \( \mathbf{E} X_n\) as \(n \rightarrow \infty\) ?

[Pittman [236, #18]