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Unformatted text preview: ISyE 2027, Spring 2010 HW 9 Solution 1. Let Y = 1 + μX , or X = Y 1 μ . F Y ( y ) = Pr ( Y ≤ y ) = Pr ( X ≤ y − 1 μ ) = F X ( y − 1 μ ) . f Y ( y ) = dF X ( y 1 μ ) dy = dF X ( x ) dx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = y 1 μ dx dy = f X ( x ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = y 1 μ dx dy . The pdf of Y , f Y ( y ), becomes f Y ( y ) = f X ( x ) dx dy = 2 e 2 x 1 μ , x ≥ , = 2 μ e 2( y 1) μ , y ≥ 1 , and zero otherwise. 2. Let Y = 1 + 2 √ Z , or Z = ( Y 1) 2 4 . Y is one of 1 , 3 , 1 + 2 √ 2 , 1 + 2 √ 3 , and 5, since Z is one of , 1 , 2 , 3 , and 4. The pmf of Y , p Y ( y ), becomes p Y ( y ) = Pr ( Y = y ) = Pr (1 + 2 √ Z = y ) = Pr ( Z = ( y − 1) 2 4 ) = ( 4 ( y 1) 2 4 )( 1 4 ) ( y 1) 2 4 ( 3 4 ) 4 ( y 1) 2 4 , y = 1 , 3 , 1 + 2 √ 2 , 1 + 2 √ 3 , 5 , , otherwise. 3. Let Y = sin ( U ). The distribution function of Y , F Y ( y ), becomes F Y ( y ) = Pr ( Y ≤ y ) = Pr (sin( U ) ≤ y ) = Pr ( U ≤ a 1 ) + Pr ( U ≥ a 2 ) , where a 1 and a 2 are such that sin...
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This note was uploaded on 02/21/2011 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Tech.
 Fall '08
 Zahrn

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