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

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Unformatted text preview: STAT 410 Recall: Example 8: 2x 0 Examples for 01/28/2008 Spring 2008 fX( x) = Y= X. 0 < x <1 o.w. FX( x ) = x2 1 0 x<0 0 ≤ x <1 x ≥1 y<0 y≥0 F Y ( y ) = P ( Y ≤ y ) = P ( X ≤ 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. F Y ( y ) = F X ( y 2 ) = 1. 0 4 y≥1 FY( y ) = y 1 - y<0 0 ≤ y <1 y ≥1 - fY( y) = - 4 y3 0 - 0 < y <1 o.w. - 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 [ g – 1( y ) ] dx /d y = /d y dx dy dx 2y - f Y ( y ) = f X ( g – 1( y ) ) - g( x) = x g – 1( y ) = y 2 dx dy = ( 2 y2 ) ( 2 y ) = 4 y3 /d y = f Y ( y ) = f X ( g – 1( y ) ) 0<y<1 Example 9: fX( x) = Y = 1 X2. 6 x5 0 0 < x <1 o.w. / g ( x ) = 1/ x 2 dx g – 1( y ) = 1 y 1 = y – /2 /d y = – 1 – 3 /2 y 2 dx dy f Y ( y ) = f X ( g – 1( y ) ) 1 5 3 = ( 6 y – /2 ) ( y – /2 ) = 3 y – 4 2 y>1 Example 10: Z ~ Standard Normal X = Z2 FX( x ) = P( X ≤ x ) = P( Z2 ≤ x ) = P(− = − x ≤Z≤ x) x x - 1 2 - e − z2 2 dz d dx β (x ) α (x ) - - - - - - - - - - f ( x, z ) d z = β ' ( x ) ⋅ f ( x, β ( x )) β (x ) α (x ) + ∂ [ f ( x, z ) ] d z ∂x – α ' ( x ) ⋅ f ( x,α ( x )) - - - - - - - - - - - - - - fX( x) = 1 2 1 x 2 = = e −x 2 −− 1 2 1 x e −x 2 2 e −x 2 1 21 2 x −1 2 1 Γ (1 2 ) 2 12 x (1 2) − 1 e −x 2 , x > 0. X~ χ 2( 1) OR ∞ −∞ MX( t ) = E( et Z ) = ∞ −∞ 2 e t z2 1 2 e − z2 2 dz 1 = 1 2 e − z 2 2 ⋅ (1 − 2 t ) ( ) dz = (1 − 2 t ) 12 , t < 1/ 2, 1 ) random variable. 1− 2t since ( 1 − 2 t )1 2 e − (z 2 2 X has a 2 )⋅ (1 − 2 t ) is the p.d.f. of a N ( 0, χ2( 1 ) distribution. ...
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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

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