(a) Give the definition of a Poisson process $\left(N_{t}, t \geqslant 0\right)$ with rate $\lambda$, using its transition rates. Show that for each $t \geqslant 0$, the distribution of $N_{t}$ is Poisson with a parameter to be specified.

Let $J_{0}=0$ and let $J_{1}, J_{2}, \ldots$ denote the jump times of $\left(N_{t}, t \geqslant 0\right)$. What is the distribution of $\left(J_{n+1}-J_{n}, n \geqslant 0\right) ?$ (You do not need to justify your answer.)

(b) Let $n \geqslant 1$. Compute the joint probability density function of $\left(J_{1}, J_{2}, \ldots, J_{n}\right)$ given $\left\{N_{t}=n\right\}$. Deduce that, given $\left\{N_{t}=n\right\},\left(J_{1}, \ldots, J_{n}\right)$ has the same distribution as the nondecreasing rearrangement of $n$ independent uniform random variables on $[0, t]$.

(c) Starting from time 0, passengers arrive on platform $9 \mathrm{~B}$ at King's Cross station, with constant rate $\lambda>0$, in order to catch a train due to depart at time $t>0$. Using the above results, or otherwise, find the expected total time waited by all passengers (the sum of all passengers' waiting times).