quiz4_Prac - -2 β-α(1-β-2 α 3 Textbook page 235 4.4.20...

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ST561: Study problems for Quiz #4 1. Let Y 1 and Y 2 have joint density function f Y 1 ,Y 2 ( y 1 , y 2 ) = 8 y 1 y 2 , 0 < y 1 < y 2 < 1 . and U 1 = Y 1 /Y 2 and U 2 = Y 2 . Derive the joint density of ( U 1 , U 2 ). Show that U 1 and U 2 are independent. 2. Let X follow a gamma distribution with parameters α > 0 and β > 0. Let Y = e X , which is called log-gamma distribution. The log-gamma distribution is used by actuaries as part of an important model for the distribution of insurance claims. Find the pdf of Y . If β < 1, show that E ( Y ) = (1 - β ) - α . If β < 0 . 5, show that V ( Y ) = (1
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Unformatted text preview: -2 β )-α-(1-β )-2 α . 3. Textbook page 235, 4.4.20 4. Let Z 1 ,Z 2 ,...,Z 5 be i.i.d. N (0 , 1). Let Z = 1 5 ∑ 5 i =1 Z i . Let Z 6 be another independent observation from the same population N (0 , 1). What is the distribution of • X 1 = ∑ 5 i =1 Z 2 i . • X 2 = ∑ 5 i =1 ( Z i-Z ) 2 + Z 2 6 . • X 3 = √ 5 Z 6 / q ∑ 5 i =1 Z 2 i . • X 4 = 2(5 Z 2 + Z 2 6 ) / ∑ 5 i =1 ( Z i-Z ) 2 . 1...
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