(a) A fire station is to be located along a road of length A, A < ∞. If fires occur at points uniformly chosen on (0, A), where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to
minimize E[|X − a|]
when X is uniformly distributed over (0, A).
(b) Now suppose that the road is of infinite length—stretching from point 0 outward to ∞. If the distance of a fire from point 0 is exponentially distributed with rate λ, where should the fire station now be located? That is, we want to minimize E[|X − a|], where X is now exponential with rate λ.