Question

The county hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangu- lar, so the travel distance from the hospital, whose coordinates are (0, 0), to the point (x, y) is |x| + |y|. If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance.

Answer 1

3

Explanation:

The x-coordinate of the accident is a random variable of uniform distribution with range [-1.5,1.5]. The same can be said for the y-coordinate. However, |x| and |y| are also uniform variables with range [0,1.5], since we still have the same probability on every pair of intervals with the same lenght.

Note that |X| and |Y| are independent with each other, therefore E(|x|+|y|) = E(|x|) + E(|y|) = 2*E(|x|). The last equality holds because both x and y are identically distributed, then so are |x| and |y|.

E(|x|) =
`\int\limits^(1.5)_0 {(x)/(1.5) \, dx = (2)/(3)* ((1.5^2)/2 - 0^2/2) = (3)/(2)`

Therefore, E(|x| + |y|) = 2 * 3/2 = 3.

I hope that works for you!