Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 48 / 5 / b / Solution

The event that the first arrival exceeds $t$ is exactly the event $N_t=0$. Thus $\mathbb{P}({\tau }_1>t)=e^{-\lambda t}$ for $t\ge 0$, and differentiation gives the exponential distribution density $\boxed{f_{{\tau }_1}(t)=\lambda e^{-\lambda t}}$ for $t>0$.

Integration by parts, or the gamma integral, yields $\mathbb{E}{\tau }_1={\int }_0^{\infty }{e}^{-\lambda t}dt=1/\lambda$ and $\mathbb{E}{\tau }_1^2=2{\int }_0^{\infty }t{e}^{-\lambda t}dt=2/{\lambda }^2$. Therefore $\boxed{\mathbb{E}{\tau }_1=1/\lambda ,\mathrm{Var}{\tau }_1=1/{\lambda }^2}$.