Solution
= Solution
The distribution is the <geometric distribution> on $\{1,2,\ldots\}$ with success parameter $q=e^{-\lambda t}$. Its <expected value> is therefore
$$
\mathbb E[N(t)]=\frac1q=e^{\lambda t}.
$$
Equivalently, multiplying the <master equation> by $n$ and summing gives $m'(t)=\lambda m(t)$ with $m(0)=1$.