Example 6.3 (A Birth Process with Linear Birth Rate) Consider a population whose members can give birth to new members but cannot die. If each member acts independently of the others and takes an exponentially distributed amount of time, with mean 1/λ, to give birth, then if X(t) is the population size at time t, then {X(t), t  0} is a pure birth process with λn = nλ, n  0. This follows since if the population consists of n persons and each gives birth at an exponential rate λ, then the total rate at which births occur is nλ. This pure birth process is known as a Yule process after G. Yule, who used it in his mathematical theory of evolution.