In Exercise 5.38, we assumed that Y1, the weight of a bulk

QUESTION:

Problem 137E

In Exercise 5.38, we assumed that Y1, the weight of a bulk item stocked by a supplier, had a uniform distribution over the interval (0, 1). The random variable Y2 denoted the weight of the item sold and was assumed to have a uniform distribution over the interval (0, y1), where y1 was a specific value of Y1. If the supplier stocked 3/4 ton, what amount could be expected to be sold during the week?

Reference

Let Y1 denote the weight (in tons) of a bulk item stocked by a supplier at the beginning of a week and suppose that Y1 has a uniform distribution over the interval 0 ≤ y1 ≤ 1. Let Y2 denote the amount (by weight) of this item sold by the supplier during the week and suppose that Y2 has a uniform distribution over the interval 0 ≤ y2 ≤ y1, where y1 is a specific value of Y1.

a Find the joint density function for Y1 and Y2.

b If the supplier stocks a half-ton of the item, what is the probability that she sells more than a quarter-ton?

c If it is known that the supplier sold a quarter-ton of the item, what is the probability that she had stocked more than a half-ton?