Question

A fire station is to be located along a road of length A, A < q. 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).

Answer 1

Answer:

Step-by-step explanation:

Given that X is uniform in the interval (0,A)

X is continuous since it represents the distance

A fire station is to be located along a road of length A, A < q. If fires occur at points uniformly chosen on (0, A), we  should find where the station be located so as to minimize the expected distance from the fire

Distance can be taken as absolute values here as either side is the same.

E(|x-a|] is to be minimum

E(|x-a|)=E(x-a) for 0<x<a

         =E(a-x), for a<x<A

Using integral we find this value

$E(|x-a|)=\int\limits^a_0 {x-a} \, dx +=\int\limits^A_a -{x-a} \, dx\\f(a)=\frac{a^2}{2} -\frac{(A-a)^2}{2}$

$f(a)=\frac{a^2}{2} +\frac{(A-a)^2}{2} \\f'(a) = a-(A-a)\\f"(a) =2$

Using calculus we find that f" is positive so when f' =0 we get solution

f'(a) =0 gives

$a=\frac{A}{2}$

Hence fire station to be located at the mid point of 0 and A