010+Correlation

The joint probability distribution and the joint

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Unformatted text preview: dx · = (y − µy )fy (y )dy = 0. The joint probability distribution and the joint probability density function can be characterized by means, variances and covariance: +∞ µx = E (x) = xP (X = x) = xfx (x)dx, −∞ x∈X +∞ µy = E (y ) = yfy (y )dy, yP (Y = y ) = −∞ y ∈Y 2 σx = V ar(x) = E (x − µx )2 = (x − µx )2 P (X = x) = x∈X +∞ = (x − µx )2 fx (x)dx, −∞ 2 σy = V ar(y ) = E (y − µy )2 = (y − µy )2 P (X = y ) = y ∈X +∞ = −∞ σxy = CoV ar(x, y ) = E [(x − µx )(y − µy )] = 2 (y − µy )2 fy (y )dy, A. Zhensykbaev Correlation = (x − µx )(y − µy ) = x∈X y ∈Y = (x − µx )(y − µy )f (x, y )dxdy. Covariance matrix is 2 σx σxy 2 σxy σy Σ= Unbiased estimators are µx = x, ˆ ¯ 1 σx = ˆ n−1 µy = y , ˆ ¯ n 1 σy = ˆ n−1 2 (xi − x) , ¯ i=1 σxy ˆ 1 = n−1 n (yi − y )2 ¯ i=1 n (xi − x)(yi − y ). ¯ ¯ i=1 Correlation coecient. The stochastic dependence of X on Y or Y on X is characterized in some sense by the correlation coecient ρ= σxy 2 σx · 2 σy = E [(x − E (x))(y ...
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