An Island is initially uninhabitated. Creatures arrive as a poisson process of rate a per month. Once there each creature produces offspring as a poisson process of rate b per month. Assume that the Poisson process for arrivals and births are all independent. Assume also that there are no deaths and no creatures leave the island. let X(t) be the number of creatures on the island at time t months and $\displaystyle P_{n}(t)=P(X(t)=n)$.

What is the probability that the island is inhabitated by time 3 months?

Do they mean there is at least one creature after 3 months? I have a formula for $\displaystyle P_{n}(t)$ but I am not sure what is n=? when t=3.

thanks for any help.