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01_28 - STAT 410 Recall Example 8 2x 0 Examples for Spring...

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STAT 410 Examples for 01/28/2008 Spring 2008 Recall : Example 8 : f X ( x ) = ° ± ² < < o.w. 0 1 0 2 x x F X ( x ) = ³ ° ³ ± ² < < 1 1 1 0 0 0 2 x x x x Y = X . y < 0 F Y ( y ) = P ( Y y ) = P ( X y ) = 0. y 0 F Y ( y ) = P ( Y y ) = P ( X y ) = P ( X y 2 ) = F X ( y 2 ) . 0 y < 1 F Y ( y ) = F X ( y 2 ) = y 4 . y 1 F Y ( y ) = F X ( y 2 ) = 1. F Y ( y ) = ³ ° ³ ± ² < < 1 1 1 0 0 0 4 y y y y f Y ( y ) = ³ ° ³ ± ² < < o.w. 0 1 0 4 3 y y - - - - - - - - - - - - - - Theorem 1.7.1 X – continuous r.v. with p.d.f. f X ( x ) . Y = g ( X ) g ( x ) – one-to-one, differentiable d x / d y = d [ g 1 ( y ) ] / d y ´ f Y ( y ) = f X ( g 1 ( y ) ) y x d d - - - - - - - - - - - - - - g ( x ) = x g 1 ( y ) = y 2 d x / d y = 2 y f Y ( y ) = f X ( g 1 ( y ) ) y x d d = ( 2 y 2 ) ( 2 y ) = 4 y 3 0 < y < 1
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Example 9 : f X ( x ) = ³ ° ³ ± ² < < o.w. 0 1 0 6 5 x x Y = 1 / X 2 . g ( x ) = 1 / x 2 g 1 ( y ) = y 1 = y 1 / 2 d x / d y = – 2 1 y 3 / 2 f Y ( y ) = f X ( g 1 ( y ) ) y x d d = ( 6 y 5 / 2 ) ( 2 1 y 3 / 2 ) = 3 y 4 y > 1 Example 10 : Z ~ Standard Normal
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