For two random variates and , the correlation is defined bY
(1)
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where denotes standard deviation and is the covariance of these two variables. For the general case of variables and , where , 2, ..., ,
(2)
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where are elements of the covariance matrix. In general, a correlation gives the strength of the relationship between variables. For ,
(3)
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The variance of any quantity is always nonnegative by definition, so
(4)
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From a property of variances, the sum can be expanded
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(6)
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(7)
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Therefore,
(8)
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Similarly,
(9)
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(10)
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(11)
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(12)
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Therefore,
(13)
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so .
For a linear combination of two variables,
(14)
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(15)
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(16)
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(17)
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Examine the cases where ,
(18)
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(19)
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The variance will be zero if , which requires that the argument of the variance is a constant. Therefore, , so . If , is either perfectly correlated () or perfectly anticorrelated () with .