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010+Correlation

# Thus 0 but z and x are dependent 4 a zhensykbaev

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Unformatted text preview: = E [(Z − E (Z ))(X − E (X ))] V ar(Z ) · V ar(X ) , Since E (X ) = 0 we have E (Z ) = E (XY ) = E (X ) · E (Y ) = 0, and in view of E (Y ) = 0 E [(Z − E (Z ))(X − E (X ))] = E (ZX ) = E (X 2 Y = E (X 2 ) · E (Y ) = 0. Thus, ρ = 0, but Z and X are dependent. 4 A. Zhensykbaev Correlation Pearson's empirical or sample correlation coecient. Let us given a sample of n pairs (xi , yi ), which are independent and identically distributed with the probability density function f (x, y ). The estimator of the correlation coecient ρ (the theoretical moment of f (x, y )) is the sample correlation coecient (or empirical moment) ρ= ˆ SSxy . S Sxx SSyy Test of hypothesis about a correlation coecient. Let random vectors (xi , yi ) are normally distributed. Assume that the null hypothesis is H0 = {ρ = 0}, alternative hypothesis is Ha = {ρ = 0}. Test statistic is ρ2 . ˆ Rejection region is RR = ρ2 ≥ ˆ 1 1 + (n − 2)/t2 2 α/ , where t2 2 is dened from the Table VI (McClave and Benson) for the α/ degree of...
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