20101ee131A_1_ee131A_1_hw7sol-2010

20101ee131A_1_ee131A_1_hw7sol-2010 - 4.99 4.103 b) x x x x...

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4.99 4.103
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b) 22 00 0 0 2 0 11 () 2 ( ) 2 (1 ) 1 23 3 xx x x yy x x PY X e dx ye dy e dx e ee d x ∞∞ −− <= = =− = = ∫∫ 7.7 a) 21 0 2 0 () () 2 1 0 ( 2) ( 1 0) ZX Y j jj j ωω ω Φ= ΦΦ= = b) 20 5 1 1 (2 )(10 ) 2 2 10 52 0 42 41 0 Z j j ⎛⎞ = ⎜⎟ ⎝⎠ Thus, by the linearity of Fourier transform, 0 2 1 0 51 5 5 2 1 0 , 0 44 2 2 zz z z Z fz e e e e z =×× −×× = .
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7.8 a) 37 3 7 () ( ) ( )( ) ( 3) (7) jX jY ZX Y Ee e ωω ω −− Φ= = = Φ Φ b) Notice that (0) (0) 1 XY , 00 0 11 1 ( 3) (7) 1 1 3( 3 )7( 7 )3( 0 ) 7( 0 ) 3( ) 7() Y X Y EZ jj j j j EX EY == = ∂∂ ′′ −Φ =− 2 2 0 1 ( ) Z j = Since ( ) (3 ) ( 7 ) 3 (3 ) ( 7 ) 7 (3 ) ( 7 ) Y X Y X Y Φ = ΦΦ −= , 22 0 0 2 ( ) (3 ) ( 7 ) 3 (3 ) ( 7 ) 7 (3 ) ( 7 ) 3 ( 3) (7) 7 ( 1 9 (0) 21 (0) (0) 21 (0) (0) 49 (0) 9( ) 4 2( )() 4 9( ) Y X Y X Y Y X Y XX YX YY j EXEY = = Φ = Φ = Φ Φ Φ Φ + Φ + Thus, 2 2 ( ) 9( ) 9( ) 4 9() 4 2( )() 9( ) 4 ) Var Z E Z E Z EX EY Var X Var Y + + =+
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7.22 a) 100 100 100 100 100 0.5 40 100 0.5 60 100 0.5 (40 60) 0.5 100 0.5 100 0.5 100 (2 ) ( 2 ) 1 2 ( 2 ) 100 0.5 50 100 0.5 55 100 0.5 (50 55) 0.5 100 0.5 100 0.5 100 0.5 (1) S PS P QQ Q S P Q −× ⎛⎞ ≤≤= ⎜⎟ ××× ⎝⎠ ≈− = b) 1000 1000 1000 1000 1000 0.5 400 1000 0.5 600 1000 0.5 (400 600) 0.5 1000 0.5 1000 0.5 1000 ( 6.3) (6.32) 1 2 (6.32) 1 1000 0.5 500 1000 0.5 550 1000 0.5 (500 550) 0.5 1000 0.5 1000 0.5
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This note was uploaded on 09/28/2010 for the course EE 131A EE 131A taught by Professor Vwaniroychowdhury during the Winter '10 term at UCLA.

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20101ee131A_1_ee131A_1_hw7sol-2010 - 4.99 4.103 b) x x x x...

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