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

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.

Reference

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

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.

Solution :

Step 1 of 2:

The proportion of the capacity sold during the week is given by

Our goal is:

a). We need find .

a). We need find .

Now we have to find F(1/2, 1/3) = .

We know that F(a,b) = .

F(a,b)=

Here F(,) =

F(,) =

F(,) =

F(,) =

F(,) =

F(,) =

F(,) =

F(,) =

F(,) =

Therefore is 0.1064.