Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 6B / iii / Solution

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 $m(0)=1$.