Covariance
Covariance
Covariance
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<strong>Covariance</strong><br />
<strong>Covariance</strong>: W = (X , µ X )(Y , µ Y ),<br />
Cov [XY ] = E[W ] = E[(X , µ X )(Y , µ Y )]<br />
<strong>Covariance</strong> is also<br />
Cov [XY ] = E[XY],µ X µ Y<br />
Cov [XY ] > 0says,X > E[X] implies Y > E[Y ]<br />
is likely (X goes up, Y goes up)<br />
Var[X +Y ] = Var[X] + Var[Y ] + 2Cov[XY ]<br />
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Correlation<br />
The correlation of X and Y is E[XY]<br />
Correlation = <strong>Covariance</strong> is E[X] = E[Y] = 0<br />
E[XY] > 0 suggests that X > 0 increases chance<br />
Y > 0<br />
Orthogonal: E[XY] = 0<br />
Uncorrelated: Cov[XY] = 0<br />
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Correlation Coefficient<br />
The correlation coefficient of two random<br />
variables X and Y is<br />
ρ X Y =<br />
Cov [X Y ]<br />
p<br />
Var[X]Var[Y ]<br />
Thm: the Correlation coefficient is normalized:<br />
,1 ρ X Y 1<br />
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Thm: If Y = aX + b, then<br />
ρ X Y =<br />
8<br />
><<br />
>:<br />
,1 a < 0<br />
0 a = 0<br />
1 a > 0<br />
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