This is quite an interesting problem, but I'm not sure how to go about doing it. I know that by using some basic Poisson properties I can figure it out but I'm failing to see how. It goes like this:
A population comprises of $X_n$ individuals at time $n=1,2,3...$ Suppose that $X_0$ has Poisson ($\mu$) distribution. Between time $n$ and time $n+1$ each of the $X_n$ individuals dies with probability $p$, independently of the others. The population at time $n+1$ is formed from the survivors together with a random number of immigrants who arrive independently according to a Poisson ($\mu$) distribution. What is the distribution of $X_n$? I feel that I'm close but can't quite get it. It sounds like I could set up a series and then maybe prove it by induction.