91Ó°ÊÓ

A rook and a bishop perform independent symmetric random walks with synchronous steps on a \(4 \times 4\) chessboard ( 16 squares). If they start together at a corner, show that the expected number of steps until they meet again at the same comer is \(448 / 3\).

Let \(X\) be a \(n\)-dimensional Poisson process with constant intensity \(\lambda\). Show that the volume of the largest \((n\)-dimensional) sphere centred at the origin which contains no point of \(X\) is exponentially distributed. Deduce the density function of the distance \(R\) from the origin to the nearest point of \(X\). Show that \(E(R)=\Gamma(1 / n) /\left(n(\lambda c)^{1 / n}\right)\) where \(c\) is the volume of the unit ball of \(R^{n}\) and \(\Gamma\) is the gamma function.

Parrando's paradox. A counter performs an irreducible random walk on the vertices \(0,1,2\) of the triangle in the figure beneath, with transition matrix $$ \mathbf{P}=\left(\begin{array}{ccc} 0 & p_{0} & q_{0} \\ q_{1} & 0 & p_{1} \\ p_{2} & q_{2} & 0 \end{array}\right) $$ where \(p_{i}+q_{i}=1\) for all \(i\). Show that the stationary distribution \(\pi\) has $$ \pi_{0}=\frac{1-q_{2} p_{1}}{3-q_{1} p_{0}-q_{2} p_{1}-q_{0} p_{2}} $$ with corresponding formulae for \(\pi_{1}, \pi_{2}\),

Cars arrive at the beginning of a long road in a Poisson stream of rate \(\lambda\) from time \(t=0\) onwards. A car has a fixed velocity \(V>0\) which is a random variable. The velocities of cars are independent and identically distributed, and independent of the arrival process. Cars can overtake each other freely. Show that the number of cars on the first \(x\) miles of the road at time \(t\) has the Poisson distribution with parameter \(\lambda \mathrm{E}\left[\boldsymbol{V}^{-1} \min (x, V t)\right]\).