Two towns are connected by a road and a railway through a valley. a wildfire closes the railway but not the road, a snowstorm closes the road but not the railway, and an earthquake closes both. wildfires, snowstorms and earthquakes occur at times of three independent poisson processes, rate 11 for wildfires, rate 12 for snowstorms, and rate 13 for earthquakes. let u be the time till the next railway closure and v the time till the next road closure.
(a) find the probability p(u> ,v> t). hint: let n; denote a poisson process of rate li. express the event in terms of n1, n2 and n3.
(b) find the distribution of u and the distribution of v.
(c) determine whether u and v are independent. hint: check whether p(u > s, v > t) equals p(u > s). p(v > t).