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rjawbreakerca 2022-07-07 Answered
Now when we take Z 1 = X + Y 1 and Z 2 = X + Y 2 , what can we say about the correlation coefficient between Z 1 and Z 2 ?
For this case, is it possible to find the correlation coefficient as function of σ x and σ y ?
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Answers (1)

Tamia Padilla
Answered 2022-07-08 Author has 16 answers
Step 1
By definition of correlation,
corr ( Z 1 , Z 2 ) Cov ( Z 1 , Z 2 ) Var ( Z 1 ) Var ( Z 2 ) .
The numerator is
Cov ( Z 1 , Z 2 ) = Cov ( X + Y 1 , X + Y 2 ) = Var ( X ) + Cov ( Y 1 , X ) + Cov ( X , Y 2 ) + Cov ( Y 1 , Y 2 ) .
We need information on how X is correlated with Y 1 , Y 2 to proceed further, but assuming X , Y 1 , Y 2 are all uncorrelated,
Cov ( Z 1 , Z 2 ) = Var ( X ) Var ( Z 1 ) = Var ( X ) + Var ( Y 1 ) Var ( Z 2 ) = Var ( X ) + Var ( Y 2 ) corr ( Z 1 , Z 2 ) = σ x 2 σ x 2 + σ y 2 .
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