# 01_25 - STAT 410 Examples for Spring 2008 Transformations...

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Unformatted text preview: STAT 410 Examples for 01/25/2008 Spring 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 ( ) = 0.4 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, … . Example 4 : U ~ Uniform ( 0, 1 ): f U ( u ) = & ¡ ¢ < < o.w. 1 1 x F U ( u ) = £ & £ ¡ ¢ ≥ < ≤ < 1 1 1 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 ) = £ £ & £ £ ¡ ¢ < < otherwise 1 2 1 y y Example 5 : U ~ Uniform ( 0, 1 )....
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01_25 - STAT 410 Examples for Spring 2008 Transformations...

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