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ps08 - Statistics 403 Problem Set 8 Due in lab on Friday...

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Statistics 403 Problem Set 8 Due in lab on Friday, November 12th 1. Suppose X 1 , . . . , X n and Y 1 , . . . , Y n are both iid samples, with EX i = EY i = 0, var( X i ) = σ 2 X , var( Y i ) = σ 2 Y , cor( X i , Y i ) = r , and cor( X i , Y j ) = 0 if i 6 = j . (a) What is cov( X 1 , i X i )? Solution: cov( X 1 , X i X i ) = X i cov( X 1 , X i ) = cov( X 1 , X 1 ) + 0 + · · · + 0 = σ 2 X . (b) What is cor( X 1 , i X i )? Solution: cor( X 1 , X i X i ) = cov( X 1 , i X i ) SD( X 1 )SD( i X i ) = σ 2 X σ X q 2 X = 1 / n. (c) What is cov( X 1 , ¯ X )? Solution: cov( X 1 , ¯ X ) = cov( X 1 , X X i /n ) = n - 1 cov( X 1 , X i X i ) = σ 2 X /n (d) What is cor( X 1 , ¯ X )? 1
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Solution: cor( X 1 , ¯ X ) = cov( X 1 , ¯ X ) SD( X 1 )SD( ¯ X ) = σ 2 X /n σ X · σ X / n = 1 / n. (e) What is cov( i X i , i Y i )? Solution: cov( X i X i , X i Y j ) = X i,j cov( X i , Y j ) = X i cov( X i , Y i ) = nrσ X σ Y . (f) What is cor( i X i , i Y i )? Solution: cor( X i X i , X i Y i ) = cov( i X i , i Y i ) SD( i X i )SD( i Y i ) = nrσ X σ Y q 2 X q 2 Y = nrσ X σ Y X σ Y = r. (g) What is cov( ¯ X, ¯ Y )? Solution: cov( ¯ X, ¯ Y ) = cov( n - 1 X i X i , n - 1 X i Y i ) = n - 2 cov( X i X i , X i Y i ) 2
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= n - 2 · nrσ X σ Y = n - 1 X σ Y .
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