Chapter 5.2, Problem 5E

Refer to Example 5.4. The joint density of Y₁, the proportion of the capacity of the tank that is stocked at the beginning of the week, and Y₂, the proportion of the capacity sold during the week, is given by

$f(y_1,y_2)=\{\begin{array}{ll}3y_1,\hfill & 0\le y_2\le y_1\le 1,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

a Find F(1/2, 1/3) = P(Y₁ ≤ 1/2, Y₂ ≤ 1/3).

b Find P(Y₂ ≤ Y₁/2), the probability that the amount sold is less than half the amount purchased.

EXAMPLE 5.4

Gasoline is to be stocked in a bulk tank once at the beginning of each week and then sold to individual customers. Let Y₁ denote the proportion of the capacity of the bulk tank that is available after the tank is stocked at the beginning of the week. Because of the limited supplies, Y₁ varies from week to week. Let Y₂ denote the proportion of the capacity of the bulk tank that is sold during the week. Because Y₁ and Y₂ are both proportions, both variables take on values between 0 and 1. Further, the amount sold, y₂, cannot exceed the amount available, y₁. Suppose that the joint density function for Y₁ and Y₂ is given by

$f(y_1,y_2)=\{\begin{array}{ll}3y_1,\hfill & 0\le y_2\le y_1\le 1,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

A sketch of this function is given in Figure 5.4.

Find the probability that less than one-half of the tank will be stocked and more than one-quarter of the tank will be sold.