Question

PROBLEM 2 (13 points) 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, its distance from one of the fixed ends of the road is uniformly distributed over (0, L).) Assuming that the ambulance's location at the moment of the accident is also uniformly distributed, compute, assuming independence the distribution of its distance from the accident. Solution Let X denote the position of the ambulance and Y denote the position of the accident. Both X and Y are uniformly distributed over (0, L). Let Z = X-YL Then we have the following 2 2z 1 L-z 2L Thus to find the pdf fz(z) of Z, we can differentiate Fz(z) to obtain: fz(z) 2(L- z) dFz(z) otherwise This can also be proved graphically as follows.

In the above problem, what are the natural places where you would put the hospital? Why? Now, strengthen your intuition by determining the expected distance between the random accident point and the fixed point a where the hospital is located. Find the value of for which this expectation is minimized. Then look at your first, intuitive answer. What have you learned?

Answer 1

You can find the required solution in text bookhelp also.

Text book help > A first Course inprobability(7th) > chapter 6 > 14E

Let X and Y are the locations of the ambulance and the accident of the moment the accident occurs. X and Y are independent and follows uniform distribution The joint density function of f(x, y)-f(x)·f(y) The distance from the accident is Y-X1 L-a yta L-a y where 0 <a <L

Let X and Y are the locations of the ambulance and the accident of the moment the accident occurs. X and Y are independent and follows uniform distribution The joint density function of f(x, y)-f(x)·f(y) The distance from the accident is Y-X1 L-a yta L-a y where 0 <a <L