Problems

When he sets out on a drive Mr Toad selects a speed \(V\) kilometres per minute where \(V\) is a random variable with probability density \[ \alpha v^{-2}\mathrm{e}^{-\alpha v^{-1}} \] and \(\alpha\) is a strictly positive constant. He then drives at constant speed, regardless of other drivers, road conditions and the Highway Code. The traffic lights at the Wild Wood cross-roads change from red to green when Mr Toad is exactly 1 kilometre away in his journey towards them. If the traffic light is green for \(g\) minutes, then red for \(r\) minutes, then green for \(g\) minutes, and so on, show that the probability that he passes them after \(n(g+r)\) minutes but before \(n(g+r)+g\) minutes, where \(n\) is a positive integer, is \[ \mathrm{e}^{-\alpha n(g+r)}-\mathrm{e}^{-\alpha\left(n(g+r)\right)+g}. \] Find the probability \(\mathrm{P}(\alpha)\) that he passes the traffic lights when they are green. Show that \(\mathrm{P}(\alpha)\rightarrow1\) as \(\alpha\rightarrow\infty\) and, by noting that \((\mathrm{e}^{x}-1)/x\rightarrow1\) as \(x\rightarrow0\), or otherwise, show that \[ \mathrm{P}(\alpha)\rightarrow\frac{g}{r+g}\quad\mbox{ as }\alpha\rightarrow0. \] {[}NB: the traffic light show only green and red - not amber.{]}