6. 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 1 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 > s, V > t). Hint: Let N; denote a Poisson process of rate A;. Express the event in terms of M1, 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)