Chapter 5.6, Problem 82E

In Exercise 5.38, we determined that the joint density function for Y₁, the weight in tons of a bulk item stocked by a supplier, and Y₂, the weight of the item sold by the supplier, has joint density

$f(y_1,y_2)=\{\begin{array}{ll}1/y_1,\hfill & 0\le y_2\le y_1\le 1,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

In this case, the random variable Y₁ − Y₂ measures the amount of stock remaining at the end of the week, a quantity of great importance to the supplier. Find E(Y₁ − Y₂).

5.38 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.

1. a Find the joint density function for Y₁ and Y₂.
2. b If the supplier stocks a half-ton of the item, what is the probability that she sells more than a quarter-ton?
3. 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?