(i) What is meant by a Poisson process of rate $\lambda$ ? Show that if $\left(X_{t}\right)_{t \geqslant 0}$ and $\left(Y_{t}\right)_{t \geqslant 0}$ are independent Poisson processes of rates $\lambda$ and $\mu$ respectively, then $\left(X_{t}+Y_{t}\right)_{t \geqslant 0}$ is also a Poisson process, and determine its rate.

(ii) A Poisson process of rate $\lambda$ is observed by someone who believes that the first holding time is longer than all subsequent holding times. How long on average will it take before the observer is proved wrong?