Let $\lambda:[0, \infty) \rightarrow(0, \infty)$ be a continuous function. Explain what is meant by an inhomogeneous Poisson process with intensity function $\lambda$.

Let $\left(N_{t}: t \geqslant 0\right)$ be such an inhomogeneous Poisson process, and let $M_{t}=N_{g(t)}$ where $g:[0, \infty) \rightarrow[0, \infty)$ is strictly increasing, differentiable and satisfies $g(0)=0$. Show that $M$ is a homogeneous Poisson process with intensity 1 if $\Lambda(g(t))=t$ for all $t$, where $\Lambda(t)=\int_{0}^{t} \lambda(u) d u$. Deduce that $N_{t}$ has the Poisson distribution with mean $\Lambda(t)$.

Bicycles arrive at the start of a long road in the manner of a Poisson process $N=\left(N_{t}: t \geqslant 0\right)$ with constant intensity $\lambda$. The $i$ th bicycle has constant velocity $V_{i}$, where $V_{1}, V_{2}, \ldots$ are independent, identically distributed random variables, which are independent of $N$. Cyclists can overtake one another freely. Show that the number of bicycles on the first $x$ miles of the road at time $t$ has the Poisson distribution with parameter $\lambda \mathbb{E}\left(V^{-1} \min \{x, V t\}\right)$.