Consider a population of a certain colonizing species. Suppose that each individual produces offspring at a Poisson rate of λ as long as it lives, and the time until the first individual arrives is...

Consider a population of a certain colonizing species. Suppose that each individual produces offspring at a Poisson rate of λ as long as it lives, and the time until the first individual arrives is exponential with mean 1/λ. If the lifetime of an individual in the population is exponential with mean 1/µ, starting with a single individual, find the expected length of time until the population size is 5.