This exercise explores a continuous time version of a simple branching process. Consider a population of primitive organisms which do nothing but procreate and die. In particular, the population...

This exercise explores a continuous time version of a simple branching process. Consider a population of primitive organisms which do nothing but procreate and die. In particular, the population starts at time 0 with one organism. This organism has an exponentially distributed lifespan T0 with rate µ (i.e., Pr{T₀
≥ τ } = e^-µ
τ). While this organism is alive, it gives birth to new organisms according to a Poisson process of rate . Each of these new organisms, while alive, gives birth to yet other organisms. The lifespan and birthrate for each of these new organisms are independently and identically distributed to those of the first organism. All these and subsequent organisms give birth and die in the same way, again indepedently of all other organisms.