Covariance

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