Revised Example 5.5

# Revised Example 5.5 - y 2 , cannot exceed the amount...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) Example 5.5 Gasoline is to be stocked in a bulk tank once at the beginning of each week and then sold to individual customers. Let Y 1 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 supply, Y 1 varies from week to week. Let Y 2 denote the proportion of the capacity of the bulk tank that is sold during the week. Since Y 1 and Y 2 are both proportions, it is obvious that both variables take on values between 0 and 1. Further, it is obvious that the amount sold,
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Unformatted text preview: y 2 , cannot exceed the amount available y 1 . Suppose that a model for the relative frequency for Y 1 and Y 2 is given by f ( y 1 ; y 2 ) = & 3 y 1 ; & y 2 & y 1 & 1; ; elsewhere. Find the probability that less than 1 = 2 of the tank will be stocked but more than 1 = 4 will be sold. The corresponding probability is P ± Y 1 < 1 2 and Y 2 > 1 4 ² = Z 1 = 2 Z 1 1 = 4 f ( y 1 ; y 2 ) dy 2 dy 1 = Z 1 = 2 Z y 1 1 = 4 3 y 1 dy 2 dy 1 = 3 Z 1 = 2 y 1 ³ y 2 ´ y 1 1 = 4 dy 1 = 3 Z 1 = 2 ± y 1 ± 1 4 ² y 1 dy 1 = 3 4 Z 1 = 2 y 2 1 ± y 1 4 dy 1 = 3 4 µ y 3 1 3 ± y 2 1 4 ¶ 1 = 2 = ± 3 4 ²± 1 ² 1 24 ± 1 16 = 3 32 : 1...
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## This note was uploaded on 09/06/2010 for the course STAT STAT1306 taught by Professor Prof during the Fall '08 term at HKU.

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