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 coecient.
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 coecient ρ (the theoretical moment of f (x, y ))
is the sample correlation coecient (or empirical moment) ρ=
ˆ SSxy
.
S Sxx SSyy Test of hypothesis about a correlation coecient.
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 dened from the Table VI (McClave and Benson) for the
α/
degree of...
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 Spring '13
 ChristopherStocker
 Math, Correlation, Normal Distribution, Probability, Null hypothesis, Probability theory, Statistical hypothesis testing, probability density function, A. Zhensykbaev

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