7.45 Let X1 denote the amount of a certain bulk item stocked by a supplier at the beginning of a week, and suppose that X1 has a uniform distribution over the interval 0  X1  1. Let X2 denote the amount of this item sold by the supplier during the week, and suppose that X2 has a uniform distribution over the interval 0  x2  x1, where x1 is a specific value of X1.

a Find the joint density function for X1 and X2.

b If the supplier stocks an amount , what is the probability that he sells an amount greater than 1>4?

c If it is known that the supplier sold an amount equal to , what is the probability that he had stocked an amount greater than ?