 Solution

A simple birth process with parameter $\lambda$ starting from one individual is the Yule process: when its population is $n$, it waits an exponential time of rate $n\lambda$ and then moves to $n+1$.

For $f(n)=n$, the Markov jump-process generator satisfiesTherefore $\mu (t)=\mathbb{E}{X}_t$ obeysIt follows, as recorded by the mean population of a Yule process, that