Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L / a / Solution

Let $p_n(t)=\mathbb{P}(X_t=n)$. The infinitesimal definition gives the forward equationswith $p_0(0)=1$. Solving recursively givesso $X_t\sim \mathrm{Poisson}(\lambda t)$.