09_05 - STAT 410 Examples for Transformations of Random...

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STAT 410 Examples for 09/05/2008 Fall 2008 Transformations of Random Variables Example 1 : x p X ( x ) y = x 2 p Y ( y ) = p X ( y ) 1 0.2 1 0.2 2 0.4 4 0.4 3 0.3 9 0.3 4 0.1 Y = X 2 16 0.1 Example 2 : x p X ( x ) y p Y ( y ) 2 0.2 0 p X ( 0 ) = 0.4 0 0.4 4 p X ( 2 ) + p X ( 2 ) = 0.5 2 0.3 9 p X ( 3 ) = 0.1 3 0.1 Y = X 2 Example 3 : X ~ Poisson ( λ ): p X ( x ) = ! x e x - , x = 0, 1, 2, 3, 4, … . Y = X 2 p Y ( y ) = ( ) ! y e y - , y = 0, 1, 4, 9, 16, … .

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: U ~ Uniform ( 0, 1 ): f U ( u ) = ± ² < < o.w. 0 1 0 1 x F U ( u ) = ³ ³ ± ² < < 1 1 1 0 0 0 u u u u Y = U 2 F Y ( y ) = P ( Y y ) = P ( U 2 y ) y < 0 P ( U 2 y ) = 0 F Y ( y ) = 0. 0 y < 1 P ( U 2 y ) = P ( U y ) = y F Y ( y ) = y . y 1 P ( U 2 y ) = 1 F Y ( y ) = 1. f Y ( y ) = ³ ³ ³ ³ ± ² < < otherwise 0 1 0 2 1 y y Example 5 : f X ( x ) = ³ ³ ± ² < < - < < - otherwise 0 2 0 3 . 0 1
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09_05 - STAT 410 Examples for Transformations of Random...

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