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Unformatted text preview: 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 nσ 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 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 nσ 2 X q nσ 2 Y = nrσ X σ Y nσ X σ Y = r....
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This note was uploaded on 02/06/2012 for the course STAT 403 taught by Professor Kerbyshedden during the Winter '12 term at University of MichiganDearborn.
 Winter '12
 KerbyShedden
 Statistics

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