hw9sol - STAT 225 - Homework 9 - Solutions 1. 6 points Let...

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1. 6 points Let the random variable X have the following density function: f X ( x ) = αθ α x α +1 x > θ α > 0 0 otherwise Find the distribution of Y = ln X θ Y = ln ± X θ ² (i) Possible values: X > θ = X/θ > 1 = Y = ln ( X/θ ) > 0 (ii) F Y ( y ) = P ( Y y ) = P ( ln ( X/θ ) y ) = P ( X θe y ) = F X ( θe y ) = f Y ( y ) = f X ( θe y ) · θe y = αθ α ( θe y ) α +1 θe y = αe - αy y > 0 = Y exp (1 ) 2. 6 points Y = e - X (i) Possible values: X > 0 = ⇒ - X/β < 0 = 0 < Y = e - X/β < 1 (ii) F Y ( y ) = P ( Y y ) = P ( e - X/β y ) = P ( - X/β ln y ) = P ( X ≥ - β ln y ) = 1 - F X ( - β ln y ) = f Y ( y ) = - f X ( - β ln y ) · ( - β/y ) = f X ( - β ln y ) · ( β/y ) = 1 β e - 1 β ( - β ln y ) · ( β/y ) = 1 /y · e ln y = 1 /y · y = 1 0 < y < 1 = Y U (0 , 1) 3. 10 points: (a) 6 points (b) 4 points: 2 for E ( Y ) 2 for E ( Y 2 ) (a) U = Y 2 (i) Possible values: Y > 0 = U = Y 2 > 0 (ii) F U ( u ) = P ( U u ) = P ( Y 2 u ) = P ( - u Y u ) = P ( Y u ) - P ( Y ≤ - u ) | {z } 0 since Y > 0 = F
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hw9sol - STAT 225 - Homework 9 - Solutions 1. 6 points Let...

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