HW05_2011 - PROF. HONG FALL 2011 ECONOMICS 6190 PROBLEM SET...

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PROF. HONG FALL 2011 ECONOMICS 6190 PROBLEM SET # 5 Y and show that the pdf integrates to 1. (a) f X ( x ) = 1 2 e x j ; < x < 1 ; Y = j X j 3 : (b) f X ( x ) = 3 8 ( x + 1) 2 ; 1 < x < 1; Y = 1 X 2 : 2. Let X have pdf f X ( x ) = 2 9 ( x + 1) ; 1 ± x ± 2 . Find the pdf of Y = X 2 . 3. [C±B, # 2.9, p.77] If the random variable X has pdf f ( x ) = ( x 1 2 ; if 1 < x < 3 ; 0 ; otherwise Find a monotone function u ( x ) such that the random variable Y = u ( X ) has a uniform (0,1) distribution. 4. [C±B, # 2.22(a), p.79] Let X have the pdf f ( x ) = 4 3 p ± x 2 e x 2 2 ; 0 < x < 1 ;& > 0 : (a) Verify that f ( x ) is a pdf; (b) Find E ( X ) and Var ( X ) : 5. f X ( x ) = 1 2 for 0 < x < 2 : Find the pdf of Y = X (2 X ) : 6. Let X have the standard normal pdf f X ( x ) = 1 p 2 ± e x 2 = 2 for < x < 1 : (a) Find E ( X 2 ) directly, and then by using the pdf of Y = X 2 and calculating E ( Y ) . Compare the answers of both the approaches.
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HW05_2011 - PROF. HONG FALL 2011 ECONOMICS 6190 PROBLEM SET...

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