PROBLEM 4. A single fire station services an island 10 miles high by 2 miles wide. For convenience, we refer to points on the island by (x, y) coordinates, with the SW corner of the island at (0, 0), the NE corner of the island at (2, 10) and the center of the island (where the fire station is located) at (1, 5). Like Manhattan, roads on this island form a grid, so fire trucks must travel on north-south and east-west axes to get to a fire. So if a fire is at (0.7, 8.2), a truck must travel a distance 2 |1.0 0.7| + |5.0 8.2| = 3.5 miles to reach the fire from the station. (This way of computing distance is known as the Manhattan distance). Assume fires happen at locations chosen uniformly at random on the island. A fire truck is dispatched immediately from the station and travels at constant speed 30 mph to the fire. Let T be the time in minutes that the fire truck takes to arrive at the fire. (a) Compute Pr(T > 10). (b) Compute Pr(T g 2). (c) Find Pr(T S t) as a function of t, i.e., the CDF of T (Careful! Behavior of T between 2 and 10 is functionally different from below 2 and above 10. It could be worth running a quick simulation to gain intuition about the CDF). (d) Use the CDF of T to derive the PDF of T. (e) Compute the expectation Ex(T) and the variance Var(T)