Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 6B (Mathematical Biology)

This is a Yule process, a pure birth process with state-dependent birth rate ${\lambda }_n=\lambda n$. Probability flows into state $n$ from $n-1$ and leaves it for $n+1$, so the master equation is

Put $q=e^{-\lambda t}$ and propose $P_n=q(1-q{)}^{n-1}$. The initial condition follows from $q(0)=1$. Since $q^{\prime }=-\lambda q$,On the other hand,and expanding the bracket gives the same expression. The formula also sums to one by the geometric series, so it is the required probability mass function.

The distribution is the geometric distribution on $\{1,2,\dots \}$ with success parameter $q=e^{-\lambda t}$. Its expected value is thereforeEquivalently, multiplying the master equation by $n$ and summing gives $m^{\prime }(t)=\lambda m(t)$ with $m(0)=1$.