A small fruit farmer faces the decision of when to harvest his crop. The point in time at which the first freeze occurs is uniformly distributed during the month of November. (This is 30 full days, i.e., the point in time of the freeze can be any point from 0 to 30.) The farmer wants to leave the crop in the fields as long as possible, because this gives the fruit more time to mature, and the crop will fetch a better price. But if the freeze occurs before the harvest, the crop loses value.
To keep things simple, suppose the crop can be harvested at a point in time. Also, the farmer must decide before November begins when he is going to harvest (he has to reserve specialized equipment).
The crop increases in value by 100 each day it is left in the field (up to the freeze). Therefore, the farmer loses 100 for every day in between when he harvests his crop and when the freeze occurs. But if the freeze occurs before he harvests his crop, the value of the crop falls by 1000 (this loss does not depend on how much after the freeze he harvests his crop-it is a flat loss of 1000).
The optimal time to harvest would be at the exact moment of the freeze-harvesting either earlier or later involves a loss relative to the value of the crop at that optimal point in time. The farmer wants to minimize the expected value of this loss. At what point during the month of November should the farmer plan to harvest the crop?