Chapter 5
Birth and Death Processes

In a birth and death process the state is a population size, and in a short interval of length \(h\) the size can rise by one (a birth or arrival), fall by one (a death or departure), or stay the same, with probabilities proportional to \(h\). Writing down how \(P_n(t) = P(X(t) = n)\) changes over \((t, t + h]\) and letting \(h\to 0\) gives a system of differential equations, which are solved either through a generating function or, for long-run behaviour, by setting the derivatives to zero.