Chapter 5.4, Problem 49E

In Example 5.4 and Exercise 5.5, we considered 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, given by

$f\left(y_1,y_2\right)=\{\begin{array}{l}3y_1,\text{ 0}\le y_2\le y_1\le 1,\\ 0,\text{ elsewhere}\text{.}\end{array}$

Show that Y₁ and Y₂ are dependent.

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 given by

$f\left(y_1,y_2\right)=\{\begin{array}{l}3y_1,0\le y_2\le y_1\le 1,\\ 0,\text{elsewhere}\text{.}\end{array}$

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.

5.5 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

$f\left(y_1,y_2\right)=\{\begin{array}{l}3y_1,\text{ 0}\le y_2\le y_1\le 1,\\ 0,\text{ elsewhere}\text{.}\end{array}$

1. a Find F(1/2,1/3) = P(Y₁ ≤ 1/2, Y₂ ≤ 1/3).
2. b Find P(Y₂ ≤ Y₁/2), the probability that the amount sold is less than half the amount purchased.