Question

Suppose you have a very long highway which we will model as a straight line of infinite length starting at x=0 and stretching all the way to x=∞. One fire station it to be located along the road and a fire will break out at a random point along the road. The distance X of the fire from X=0 has an exponential distribution with mean equal to 4 miles.

We want to find the optimal place x=a to locate a fire station on this road. To do this we wish to minimize the function
H(a)=E[(X−a)^2],   a≥0.
Choose a to minimize this function.

Answer 1