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hw05_sol

# hw05_sol - Homework 5 Problem 1 Denote the variance as g(a...

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g ( a ) g ( a ) ( aX + (20000 - a ) Y ) = a 2 σ 2 X + (20000 - a ) 2 σ 2 Y + 2 a (20000 - a ) ρσ X σ Y dg ( a ) da = 2 2 X - 2(20000 - a ) σ 2 Y + 2(20000 - a ) ρσ X σ Y - 2 aρσ X σ Y = 0 a = 20000( σ 2 Y - ρσ X σ Y ) σ 2 X + σ 2 Y - 2 ρσ X σ Y a d 2 g ( a ) da 2 = 2( σ 2 X + σ 2 Y - 2 ρσ X σ Y ) 2( σ 2 X + σ 2 Y - 2 σ X σ Y ) = 2( σ X - σ Y ) 2 0 . E ( XY ) = R 1 0 R 1 - x - (1 - x ) 3 x 2 ydydx = R 1 0 3 x 2 2 y 2 | 1 - x - (1 - x ) dx = 0 f X ( x ) = R 1 - x - (1 - x ) 3 xdy = 6 x (1 - x ) = 6 x - 6 x 2 μ = EX = R 1 0 6 x 2 - 6 x 3 dx = 2 x 3 | 1 0 - 3 2 x 4 | 1 0 = 1 2 f Y ( y ) = R 1 -| y | 0 3 xdx = 3(1+ y 2 - 2 | y | ) 2 ν = EY = R 0 - 1 3 2 ( y + y 3 + 2 y 2 ) dy + R 1 0 3 2 ( y + y 3 - 2 y 2 ) dy = 0 Cov ( X, Y ) = E ( X · Y ) - μν = 0 - 0 = 0 X Y f X ( x ) · f Y ( y ) 6 = f X,Y ( x, y )

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E | Y | = 3 2 R 0 - 1 ( - y - y 3 - 2 y 2 ) dy + 3 2 R 1 0 ( y + y 3 - 2 y 2 ) dy = 3 R 1 0 ( y + y 3 - 2 y 2 ) dy = 1 4 E | X || Y | = = EX | Y | = R 1 0 ( - R 0 x - 1 3 x 2 ydydx + R 1 0 R 1 - x 0 3 x 2 ydy ) dx = 2 R 1 0 R 1 - x 0 3 x 2 ydydx = R 1 0 3 x 2 (1 - x ) 2 dx = 1 10 E ( X 2 ) = R 1 0 (6 x 3 - 6 x 4 ) dx = 3 10 E ( Y 2 ) = R 0 - 1 3 2 ( y 2 + y 4 + 2 y 3 ) dy + R 1 0 3 2 ( y 2 + y 4 - 2 y 3 ) dy = 2 R 1 0 3 2 ( y 2 + y 4 - 2 y 3 ) dy = 1 10 ρ | X | , | Y | = Cov ( | X | , | Y | ) σ | X | σ | Y | = E | X || Y |- E | X | E | Y | E ( X 2 ) - μ 2 E ( Y 2 ) - ( E | Y | ) 2 = 1 10 - 1 2 × 4 q ( 3 10 - 1 4 )( 1 10 - 1 16 ) = - 0 . 5774 φ X ( t ) = Z 1 0 e tx f X ( x ) dx = λ 1 - e - λ
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hw05_sol - Homework 5 Problem 1 Denote the variance as g(a...

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