hw7sol(1) - FALL 2011 ORIE 3500/5500 PROBLEM SET 7...

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FALL 2011 ORIE 3500/5500 PROBLEM SET 7 SOLUTIONS (1) Recall that ( X, Y ) have joint and marginal distributions given by x y 1 2 3 Y 1 0.217 0.153 0.057 0.427 2 0.106 0.244 0.223 0.573 X 0.323 0.397 0.28 We compute: E X =1 · 0 . 323+2 · 0 . 397+3 · 0 . 28=1 . 957 E X 2 =1 2 · 0 . 323+2 2 · 0 . 397+3 2 · 0 . 28=4 . 431 E Y =1 · 0 . 427+2 · 0 . 573=1 . 573 E Y 2 =1 2 · 0 . 427+2 2 · 0 . 573=2 . 719 E XY =1 · 1 · 0 . 217+2 · 1 · 0 . 153+3 · 1 · 0 . 057+1 · 2 · 0 . 106 +2 · 2 · 0 . 244+3 · 2 · 0 . 223=3 . 22 . (a) Cov( X, Y )= E XY - ( E X )( E Y )=3 . 22 - (1 . 957)(1 . 573)=0 . 142 . Hence, X and Y are positively correlated. (b) ρ ( X, Y )= Cov( X, Y ) radicalbig Var( X )Var( Y ) = 0 . 142 radicalbig [4 . 431 - (1 . 957) 2 ][2 . 719 - (1 . 573) 2 ] =0 . 369 . (2) We have T = 9 5 X +32 and Y = 9 5 Y +32. Hence, by the change of units formula, Cov( T, S )= parenleftbigg 9 5 parenrightbigg 2 Cov( X, Y )=9 . 72 . Since the correlation is invariant under change of units up to sign, ρ ( T, S )= ρ ( X, Y )=0 . 8 . (3) Suppose U Unif(0 , a ). We have ρ ( U, U 2 )= Cov( U, U 2 ) radicalbig Var( U )Var( U 2 ) = E U 3 - ( E U 2 )( E U ) radicalbig [ E U 2 - ( E U ) 2 ][ E U 4 - ( E U 2 ) 2 ] . 1
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It will be useful to compute the k -th moment of U : E U k = integraldisplay a 0 x k a dx = bracketleftbigg x k +1 ( k +1) a bracketrightbigg a 0 = a k k +1 .
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