Chapter 5, Problem 5.31P

a. A fire station is to be located along a road of length $A,A<\infty$. If fires occur at points uniformly chosen on $\left(0,A\right)$, where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to $\text{minimize }E\left[\left|X-a\right|\right]$ when X is uniformly distributed over (0, A).

b. Now suppose that the road is of infinite length—stretching from point 0 outward to $\infty$. If the distance of a fire from point 0 is exponentially distributed with rate $\lambda$. where should the fire station now be located? That is, we want to minimize $E\left[\left|X-a\right|\right]$ where X is now exponential with rate $\lambda$.