Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 26J / a

Let $S_1,S_2,\dots$ be independent exponential random variables of rate $\lambda$, let $T_n=S_1+\cdots +S_n$ and $T_0=0$, and define $N_t=\max \{n:T_n\le t\}$. The strong law of large numbers gives $T_n/n\to 1/\lambda$, so there is no finite-time explosion. This defines the Poisson process of rate $\lambda$.

For $n\ge 1$, repeated convolution gives $T_n$ the gamma distribution with density ${\lambda }^n{s}^{n-1}{e}^{-\lambda s}/(n-1)!$. Independence of $S_{n+1}$ givesFor $n=0$, the probability is $\mathbb{P}(S_1>t)=e^{-\lambda t}$. Hence $\boxed{N_t\sim \mathrm{Poisson}(\lambda t)}$ for all $t\ge 0$, including the degenerate law at $t=0$.