# Question: a A fire station is to be located along a

(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.

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.

## Relevant Questions

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter 1/20. Smith has a used car that he claims has been ...A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function what must the capacity of the tank be so that the ...Use the identity of Theoretical Exercise 5 to derive E[X2] when X is an exponential random variable with parameter λ. Theoretical Exercise 5 Use the result that, for a nonnegative random variable Y, to show that, for a ...Let Y = (X – ν/α)β Show that if X is a Weibull random variable with parameters ν, α, and β, then Y is an exponential random variable with parameter λ = 1 and vice versa. An ambulance travels back and forth at a constant speed along a road of length L. At a certain moment of time, an accident occurs at a point uniformly distributed on the road. [That is, the distance of the point from one of ...Post your question