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20071ee131A_1_131a8sol

# 20071ee131A_1_131a8sol - Y y = lim x →∞ F Y x,y =...

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EE 131A Homework #8 Winter 2007 Solution K. Yao 1. (3.74) Y = A cos( Wt ) + c, E [ Y ] = E [ A ]cos( Wt ) + c = m cos( Wt ) + c V ar [ Y ] = cos 2 ( Wt ) V ar [ A ] = σ 2 cos 2 ( Wt ) . 2. For 0 x 3 then 0 y 27. For other values of y , the pdf of Y is zero. For 0 y 27, dy/dx = 3 x 2 = 3 y 2 / 3 . For every value of y in 0 y 27, there is only one value of x = g - 1 ( y ) = y 1 / 3 . For 0 y 1 or 0 x 1, f ( x ) = 1 / 2, then f ( y ) = 1 6 y 2 / 3 . For 1 y 27 or 1 x 27, f ( x ) = - (1 / 4) x + (3 / 4), and f ( y ) = ( - 1 4 y 1 / 3 + 3 4 ) 1 3 y 2 / 3 . 3. (3.79) Y = K + LX, E [ Y ] = K + L E [ X ] = K + L ( n + 1) / 2 , V ar [ Y ] = V ar [ K + LX ] = V ar [ LX ] = L 2 V ar [ X ] = L 2 ( n 2 - 1) / 12 . 4. (3.80) E [ X n ] = Z 1 0 x n dx = x n + 1 n + 1 1 0 = 1 n + 1 E [ Y n ] = 1 b - a Z b a y n dy = 1 b - a " b n +1 - a n +1 n + 1 # . 5. P ( X Y 2) = Z x =2 Z x y =2 e - ( x + y ) dydx = Z x =2 ( - e - ( x + y ) | x y =2 ) dx = Z x =2 ( - e - 2 x + e - (2+ x ) ) dx = ((1 / 2) e - 2 x - e - (2+ x ) ) | x =2 = (1 / 2) e - 4 . 6. (4.5) a. All three cases have the same marginal pmf. P [ X = - 1] = P [ X = 0] = P [ X = 1] = 1 3 P [ Y = - 1] = P [ Y = 0] = P [ Y = 1] = 1 3 b. i ii iii P [ A ] 2 3 2 3 2 3 P [ B ] 5 6 2 3 2 3 P [ C ] 2 3 1 3 1 1

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7. (4.9) a. For x > 0 ,y > 0 , , then F XY ( x,y ) = Z x 0 Z y 0 axe - ax 2 / 2 bye - by 2 / 2 dxdy = (1 - e - ax 2 / 2 )(1 - e - ay 2 / 2 ) . b. P ( X > Y ) = Z 0 { Z y 0 axe - ax 2 / 2 bye - by 2 / 2 dy } dx = Z 0 axe - ax 2 / 2 (1 - e - bx 2 / 2 ) dx = 1 - a a + b . c. F X ( x ) = lim y →∞ F X ( x,y ) = 1 - e - ax 2 / 2 , x > 0 . F
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Unformatted text preview: Y ( y ) = lim x →∞ F Y ( x,y ) = 1-e-by 2 / 2 , y > . d. f X ( x ) = dF X ( x ) dx = axe-ax 2 / 2 , x > , f Y ( y ) = dF Y ( y ) dy = bye-by 2 / 2 , y > , 8. a. Let p k = P ( Y = g ( X ) = k ) , for k = 0 , 1 ,..., then p k = Z k +1 k f X ( s ) dx = Z k +1 k ae-ax dx = (1-e-a ) e-ak ,k = 0 , 1 ,... ; = 0 ,k < . b. For the Gaussian r.v. of zero mean and unit variance, we have p k = Φ( k + 1)-Φ( k ) , k ≥ 0; = Φ(-k + 1)-Φ(-k ) , k < . c. For f X ( x ) = 1 / 10 , ≤ x ≤ 10 , then p k = Z k +1 k 1 10 dx = 1 / 10 , k = 0 , 1 ,..., 9; = 0 , otherwise. 2...
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20071ee131A_1_131a8sol - Y y = lim x →∞ F Y x,y =...

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