HW9_sol_v2 - ISyE 2027, Spring 2010 HW 9 Solution 1. Let Y...

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Unformatted text preview: ISyE 2027, Spring 2010 HW 9 Solution 1. Let Y = 1 + X , or X = Y- 1 . F Y ( y ) = Pr ( Y y ) = Pr ( X y 1 ) = F X ( y 1 ) . f Y ( y ) = dF X ( y- 1 ) dy = dF X ( x ) dx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = y- 1 dx dy = f X ( x ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = y- 1 dx dy . The pdf of Y , f Y ( y ), becomes f Y ( y ) = f X ( x ) dx dy = 2 e- 2 x 1 , x , = 2 e- 2( y- 1) , 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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HW9_sol_v2 - ISyE 2027, Spring 2010 HW 9 Solution 1. Let Y...

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