A missile can travel at either a high or a low trajectory. The missile's effectiveness decreases linearly with the distance by which it misses its target, up to a distance of 2 miles at which it is...

A missile can travel at either a high or a low trajectory. The missile's effectiveness decreases linearly with the distance by which it misses its target, up to a distance of 2 miles at which it is totally ineffective. If a low trajectory is used, the missile is safe from antimissile fire. However, its accuracy is subject to the proportion of cloud cover (). Indeed the distance, d, by which is misses its target is uniformly distributed on (0,
). For the target area,
 is known to have the probability density

If a high trajectory is used, the missile will hit the target exactly, unless it is first destroyed by antimissile fire. From previous experience, the probability, S, of the missile being destroyed is thought to have the prior density
 An experiment is conducted to provide further information about S. Two missiles are launched using a high trajectory, out of which none are shot down.

(a) Give the loss incurred in having the missile miss by a distance d. (Let 0 stand for a perfect hit and 1 for a total miss.)

(b) What is the optimal Bayes trajectory?